I found out the hard way that I simply had to give up on any ambition of watching in strictly broadcast order. The biggest problems are the Australian K9 series and Torchwood: Miracle Day, whose first showing dates mesh confusingly with the other shows. But there are also cases like the animated story / webcast The Infinite Quest, which ran alongside the broadcast Season Three. And I confess that I still have not watched all of the Monster Files. (One could of course question whether a New Who rewatch should include the Torchwood audio plays, the animated stories / webcasts, the Comic Relief shorts or even the Australian K9 show. But I think that if a thing is worth doing at all - and that, of course, is a completely different issue - it's worth doing properly.)
I also misplanned the scheduling of my write-ups. For my Old Who rewatch, it worked well enough to do six stories at a time, though this got more intense towards the end as the stories got shorter. But for New Who and its spinoffs, it would have been much better to take each season, and associated mini-episodes / audios / animations, as a separate topic. As it was, I dried up about half way in, and have had to circle back at the end with individual wrap-up posts on K9, Sarah Jane and Torchwood, and now this one to finish off.
So, all that said: my basic conclusion is that the Moffat era is actually better than I remembered from first time round, and the RTD era perhaps not quite as good. The Tenth Doctor in particular will never be my favourite (admittedly, Blink is one of my favourite episodes in the whole 50 years, but he is barely in it). The first decent Tenth Doctor story is School Reunion, four episodes in; the last is The Stolen Earth/Journey's End, before the year of specials and the jumble of The End of Time. I see the Tenth Doctor as similar to the Third in some ways, with similar character flaws that don't always appeal to me.
I do like the Eleventh Doctor, Matt Smith to an extent being Tom Baker to David Tennant's Jon Pertwee; the Doctor isn't human, and is most interesting when he is most obviously not One Of Us. But I don't think he has always been well served by the show's material. Season Six's overarching narrative in particular is confusing - I still don't quite grasp the overall concept. But there are some excellent individual pieces, the standout being (of course) The Doctor's Wife. And I love the three quite different takes on the role of the companion - Amy, Amy/Rory and Clara, with Amy/Rory/Eleven being perhaps the most fun combination ever. But this is also the era of the Paternoster Gang, a hilarious and yet effective invention (and I do recommend that fans hunt out James Goss's origin story for how Vastra and Jenny got together in the first place).
One less celebrated aspect of the Moffat era is that they have cracked the art of doing short episodes, either as teasers for the next full episode or even as stories in their own right. Thus there's the hilarious fan service of Meanwhile in the Tardis II, and the great time paradoxes of Time / Space and First Night / Last Night - this is the sort of farce format at which Moffat excels, and you wouldn't want it every day, but once every year or so is absolutely fine. Of course, in general the Moffat era is darker, which I like, and I'm sympathetic to it also being more internally referential, even if that doesn't always work.
Still, my favourite New Who Doctor remains Nine, who picks the whole thing up from Rowan Atkinson parodies and Richard E. Grant caricatures, and brings it vehemently and unstoppably back to life. In my personal ranking he is second only to the Fourth Doctor and just ahead of the First. His characterisation was completely different, fresh and very watchable. He showed us that, once again, the Doctor mattered, and I have no hesitation (and quite a good track record of success) in introducing new fans to the show with Rose.
I do recommend that you give the grand watch-through a try - if you possibly can, all the way forward from 1963 to the present day, including at least the TV spinoffs (maybe skipping K9 is allowable) and Torchwood radio plays and if possible all the various short videos. There's no other way to appreciate the full variety of the show. It's fantastic.