I first tried getting into Who by renting and watching through the Key to Time season with the 4th Doctor. I made it through the first serial...and lost interest. It wasn't until the new stuff came out and I watched the 2005 season that I decided to go ALL the way back to the beginning and go from there. As a result, I now have no single favorite Doctor, and can find things to appreciate in all of them.
That said, I think it really helps to have a more modern take, even if the 2005 era is now starting to show its age as well. There are some weaker episodes, but by and large, the new stuff is simply paced in a more modern fashion. The old stuff, by comparison, is way more plodding. With new viewers, I think you really have to gauge what kind of sci-fi fan they are. Some can't appreciate an interesting idea in spite of a clunky execution. Others really need a relatable feel to the material to get past the fantastical elements. That's why I think starting in 2005 with the relaunch is the best way to go. If that gets 'em curious about the old stuff, fine and dandy, but I wouldn't hold my breath on that.
My big criticism of the new era is in Moffat's overall plotting and the quality of the writing in general since he's taken over. For whatever reason, I found RTD's treatment of Doctor Who to be generally more memorable in terms of individual episodes, and his approach to season-long arcs to be FAR more subtle. Moffat's approach is usually to add some big "tease" at the end of the episode to hint at what's coming, and it's just too...tacked on. It's not organic to the story, and seems like a gimmick. Then his execution always ends up feeling very slapdash and last-minute. He nails the emotional aspects, but if you peek behind the curtain even the slightest, you see that the story itself is a mess. And in the end, that's I think the biggest weakness of this season: the stories themselves have been relatively weak vehicles for otherwise very solid character development. I'm hoping that after Clara's gone, this improves.
Also, I LOVE Love & Monsters. But I love it because it's a one-off that shows the kind of wake the Doctor can leave behind him, for good and ill.