Background
As eventually knows, Doctor Who was canceled at the end of 1989.
Virgin Books, the publishers of the Target novelizations, had obtained the rights to write "side" books that weren't direct novelizations of TV stories, and they decided to use these rights to carry on the story of the Seventh Doctor and Ace (and, later, other companions). While they could have published the usual unambitious tie-in books that most TV and movie series go with, the editors instead decided to aim their books at an older audience, and to try to tell stories that were "too big for the small screen." After a rocky start, Virgin found their tone, and their audience, and most of the 61 NA books were deep, riveting stories that deserved to carry the mantle of Doctor Who.
Then came the TV movie, and with it an anticipated reboot of the TV series. BBC canceled Virgin's licensed (although they continued to publish New Adventures novels with the Doctor's last companion, Benny) and began publishing their own Eighth Doctor Adventures. Initially, the plan was to make them completely separate from the Virgin novels, and watered down for a more mainstream audience. But after the new show failed to materialize, and most of the Virgin writers became BBC writers, the EDAs too became serious sci-fi/fantasy rather than just hack novels.
The EDAs ended with the resurrection of the TV series in 2005, when the BBC began publishing the New Series Adventures, as mass-market tie-in books.
As excellent as the NA and EDA novels are, there are countless fans who refuse to read them. Many of these fans have read thousands of pages of forum threads, blog posts, and so on about Doctor Who. The simple fact is that Larry Miles, Paul Cornell, etc. are better writers than most of the fan base, and they're writing fiction that's meant to entertain and/or illuminate rather than forum posts meant to argue, so it's almost certain far more interesting to read them.
Why you should read them
First, many of the novels are great stories.
Alien Bodies is one of the best science fantasy books ever written; even if it weren't a Doctor Who story, it would be worth reading.
And, more to the point, they're great Doctor Who stories. There are plots, themes, and characters worth reading. There are insights into the history and workings of the Whoniverse, and into the characters you already know, and into the events you've already seen. Even if (just as with the TV show) some of those insights are later contradicted by later stories, you've still been given a new way of looking at everything you've seen. If you're really a fan of Doctor Who, why wouldn't you want that?
Finally, the novels have clearly inspired Steven Moffat. You don't have to know anything about Fitz to follow Rory's story, but you'll get more out of both stories if you've experienced both.
So, why don't people read the novels?
There's too much to read
This objection was stated by none other than Stephen Moffat. He claims that he originally followed the novels, but a dozen new full-length Doctor Who novels a year was just way too much to keep up with, compared to the 4 or so serials the TV series used to deliver.
Moffat has a point. You also don't have to read them all at once, any more than you have to watch all 26 seasons of the classic series. Read a couple of them and see what you think. If you decide it's not worth it, you've wasted a few hours of time. If you decide you want to read more, you can. Even if you decide you need to read absolutely everything, well, you've got a whole lifetime to catch up.
There's too much continuity
It's true that the novels were always more closely connected than the TV show. You can watch the rest of season 26 without having seen Battlefield and you're not missing anything except for Battlefield itself; if you read the early Benny books without having read Love and War, they won't be as good, and you may even be confused.
But again, you still don't have to read everything. Read the best and most important books, and read the summaries on the
Discontinuity Guide for the ones you skip.
The novels also have a lot more continuity references to the TV show, and you probably haven't seen all 26 seasons. But if you just watch the first episode, The Invasion of Time, Remembrance of the Daleks, The Curse of Fenric, and Survival, you've pretty much got enough to follow everything that happens in the novels. You'll miss out on some clever references and inside jokes, but who cares?
The novels are just fan fiction
I don't know what this is even supposed to mean. The novels were commercially published books sold by Virgin Books and BBC Books. By definition, they're not fan fiction. The ranges had editors who solicited writers, reviewed unsolicited submissions, and worked with the writers. And of course they're licensed by the BBC In every possible sense, they're professional fiction, not fan fiction.
Meanwhile, it's true that some of the novels were written by people who had never published professional fiction beforehand. But so what? Yes, RTD and Paul Cornell had only written for TV; Larry Miles had only published non-fiction; Mark Gatiss hadn't done any professional writing. But they turned out to be great writers. You're not picking from a random slush pile or a blog somewhere; for the most part, the novels that were published were the good ones, by writers who went on to become professional writers, if they weren't already.
Of course there are some that aren't as good as the others, but you can always skip those. (In a few cases, the plots of the weaker novels are important to later stories, but in that case there's always sites like the to give you detailed overviews of the novels you want to skip.)
But the novels aren't "real" Doctor Who
Says who? They were intended to be "real" Doctor Who. And, except for a handful of truly oddball books, most people who read them take even the worst of them as fitting into the universe, themes, and stories of the franchise. Sure, they're different stories than what you saw on TV, but then Series 7 and Season 1 are even more different, and you wouldn't claim either of them doesn't count as "real" Doctor Who. These are stories of the Doctor, his companions, his TARDIS, and his adventures in saving the universe.
But the novels didn't really happen
Who says?
RTD never said that the events of the novels were out of continuity or canon. And he certainly knew the novels, having written one of them himself, and argued at length about others, and if you watched Confidential, you saw the full set of NAs on the bookshelf in his office. When directly asked whether the novels were still in continuity, he always avoided answering. But there's a reason he had Gary Russell on staff as a continuity adviser.
Moffat, meanwhile, hasn't been afraid to answer. If you ask him whether the books are in continuity, he'll tell you that it's a stupid question, because there is no linear continuity to a show about time travelers changing the past. Everything you saw in Turn Left was on a timeline outside of normal continuity. The end of Last of the Time Lords erased everything that happened after the Toclafane arrived. The events of other stories, like Journey's End, were removed from history by the cracks in time. Hell, everything before The Big Bang happened in a different universe, before the new one was created in part out of the memories of Amy rather than actual history. Does that mean all of those stories aren't worth watching anymore?
The EDAs, like the Moffat-era TV series, were constantly changing history as they went along, and were clearly on a path toward undoing most of what happened in the second half of the series. Nobody at the time complained that they'd wasted their time reading Unnatural History because it turned out that much of what happened in that novel ended up not in the main timeline; it was worth reading because it was worth reading, not because it was source material for filling out a chronological history of the Doctor's travels. So, if the EDAs as a whole were somehow officially removed from continuity, it still wouldn't make any difference to whether they were readable.
Whether the Doctor or Clara or whoever remembers those events or not, you will remember them, just as Moffat does, and that's all that matters.
A lot of fans are resistant to the idea that history can and does change, no matter how much Moffat (and the Doctor, with Moffat's words in his mouth) tries to hammer it home. Frankly, I have no idea why those people are watching the show. But anyone who can accept Turn Left or the Cracks storyline should have no problem accepting the novels.
What about the Looms?
This one always comes up. If you don't know the story:
In one of the early NAs, Time's Crucible, we learn that in the time of Rassilon, the pre-scientific leaders of Gallifrey cursed their successors in some way. Rassilon dealt with that by replacing natural childbirth with the Looms. A couple of later novels follow up on this idea, and Lungbarrow explicitly tells us that no Time Lord except possibly the Doctor and a few other exceptions has been born naturally from Rassilon's time up to the end of that novel.
Many fans never liked the idea of the Looms. They make the Time Lords alien in exactly the wrong way. The idea of these people who sprang to life as fully formed adults and then spent decades in bizarre Gormenghast-like isolated Houses but somehow ended up as human as Drax or Rodan is just implausible to the point of silliness.
This is one of the few major plot points from the books that the TV series explicitly contradicted, and RTD even directly said that the Looms never happened as far as he's concerned.
So, does this mean the books are irrelevant? Of course not. It means that RTD has changed one element of Time Lord history that was established in the books. And that means even less when you consider that the same element of history had also been contradicted in later books. In Unnatural History, it's made pretty clear that the Doctor had 17 different and contradictory pasts, and only 1 of them involved been Loomed. If that didn't make the previous novels somehow irrelevant and unworthy of reading, why should RTD's proclamation make any difference?
Also, as silly as the Looms may have been initially, some later authors used them more interestingly, as part of making the Time Lords as alien as they were originally intended to be (before The Invasion of Time).
Finally, the Looms are hardly the only silly thing about the Whoniverse that anyone ever invented. The classic series, and even the new series, is chock full of silly ideas, some of which were ignored, others used as the germ to make more interesting stories later.
What about Human Nature, etc.?
Yes, the novel Human Nature was adapted into a different TV story, and it seems unlikely that the Doctor went through such similar events twice.
But it's hard to see why this makes a difference. Again, it doesn't matter which stories are still in the current, ever-changing continuity. Human Nature is a great novel, and having both read the book and seen the episodes makes both of them better. You don't have to pick one or the other.
If you're worried about being "spoiled", well, the stories are different enough from each other that you don't really know the story, and neither version relies too heavily on surprise anyway, but if you really want to skip this novel, nobody's stopping you; that doesn't affect the others in any way.
Why not the audios or comics or other novels instead?
This one is the best objection there is.
You've got 26 classic seasons, 7 new seasons (plus the shorts and animated episodes), the Adventure Games and other games designed to be in-universe, hundreds of books, over 100 audios, 50 years of comic strips and books, plays, etc., not to mention spinoffs like the Torchwood TV series, the Faction Paradox books, the BBV videos, and the Kaldor City audios. Why put your time into the NA and EDA novels instead of any of the other sources?
To some extent, this is just a matter of personal preference. I think the NA and EDA novels (along with a handful of Virgin Missing Adventures and Past Doctor Adventures) are the best version of Doctor Who outside of the TV series, and add the most to the TV series. Also, the fact that the novel editors and writers really believed they were carrying forward the main storyline of the franchise makes a difference.
But not everyone agrees.
Some of the Big Finish audios, in particular, occupy a ground halfway between the novels and the TV series. Besides the format being half-way to video, they're also typically half-way between the two in terms of accessibility and scope. And of course they feature the actors you know and love from TV. And Big Finish shares a lot of authors with the novels, so it's not like the stories are written by useless hacks. And they do an even better job of trying to keep their continuity together than the novel ranges did.
The comics, meanwhile, started off as childish stories that made no attempt to be consistent with anything, and progressed to typical Marvel youth-oriented fare, but they began to improve around the time of the Sixth Doctor's hiatus on TV, and after end of the classic series, the writers (including the show's final script editor Andrew Cartmel—who also wrote some NA novels) were taking them seriously as a continuation of the story of the Seventh Doctor, and produced some great stories. While they're not quite as deep or adult-oriented as the novels or audios, that also means they sometimes fit the tone of the TV series even better. The biggest problem with the comics is that they're just harder to get hold of (at least legally—I'm pretty sure you can torrent everything if you want to).
Also, the BBC animated webcasts, if you can find them, are worth watching. (If you're worried about figuring out which stories are part of continuity, Death Comes to Time should make you give up and just start enjoying yourself; if it doesn't, there's no hope for you.)
As for the various spinoffs, definitely go watch Torchwood, and the Sarah Jane Adventures, if you haven't. But skip their tie-in books. Faction Paradox, Iris Wildthyme, and the Benny New Adventures have some great books, but you'll do better to start off with their predecessors in the NA and EDA lines. For example, Faction Paradox is more fun if you've read Christmas on a Rational Planet, Alien Bodies, and Interference. And all of the rest are fun, but amateurish; it's not as easy to produce compelling video on a shoestring budget as compelling novels.
But really, try a bit of everything, and see what you enjoy most? If you disagree with me and decide to start buying all the Big Finish audios you can and skip the novels, I've got no problem with that.
What's wrong with the past-Doctor and New Series stories?
I'm guessing the Seventh and Eighth Doctors aren't your favorite, or most familiar, incarnations. And there were as many books written about the other Doctors as about them. So, why read the NAs and EDAs instead of those other books?
The first past-Doctor stories were the Virgin Missing Adventures, published as an adjunct to the NAs. They were written by many of the same writers, and some of them were pretty good (and a couple of them fit closely into the NA storylines). But the need to fit the stories into established gaps in the timeline, and to avoid doing anything memorable enough that we'd expect the Doctor and companions to have commented on the events at some point on TV, was very restrictive. Also, there was only 1/6th as much experience writing for each of the past Doctors as for the Seventh—and, unlike the NAs, many of the writers saw the MAs as a less-important diversion, a fun chance to get to play with characters they didn't really control. And of course there was no continuing storyline to build up. Finally, many of them were written with the explicit intention of retconning some troubling bit of continuity or "redeeming" some unpopular character. So, while the MAs may have been as much professionally-published fiction as the NAs, they often feel a little bit like fan fiction. The BBC Past Doctor Adventures were basically the same, only by that time the writers knew that everyone expected less from a PDA than an EDA.
The Telos novellas, on the other hand, are mostly great stories—and there aren't too many of them, and they're short—so if you want to read about your favorite past Doctors, maybe start there.
The Short Trips short story collections (both the original Virgin collections and the later Big Finish ones) also have some great stories about past Doctors; most of the books have a few real clunkers, but you can always skip over those.
Meanwhile, the New Series Adventures, on top of having all of the problems of the MA/PDA books in trying to fit into an existing story instead of building a new story, were explicitly aimed more at mass appeal. Instead of encouraging the writers to push the boundaries of what a Doctor Who story could be, they encouraged them to write the kinds of stories TV viewers would expect. Of course they ended up with many of the same writers, and some of them produced some pretty good books, but the NSAs never lived up to the standards of the NA and EDA lines.
What order?
Unless you're really ready to commit to reading 100+ novels, don't worry about that yet. Pick up a couple of the best-regarded books, maybe skipping over the ones that are too tied to a specific story arc (which, sadly, rules out some of the best ones…), and see what you think. I'd recommend Love and War, Alien Bodies, The Also People, The Scarlet Empress, Damaged Goods, Vampire Science, Just War, The Taking of Planet 5, Conundrum, Blood Heat, The Tomorrow Windows, and Sky Pirates!