Most folks have "their" Doctor. The one who caught their attention and riveted them to his episodic stories when they were younger. Moffat is no different. Check out his Time Crash mini episode from the Children in Need special he did for the BBC. It's a brilliant piece of work by David Tennant (our previous Doctor incarnation) that touches on many of the things that make the character unique and wonderful. It also serves to show WHY so many of the good Doctor's fans were disappointed when Mr. Tennant hung up his Tardis key (though with the way things work in the Doctor's world, we could run into him again).
For me Tom Baker is most frequently the one that came to mind pre- 2005 series resurrection. Though his successors "The Caves of Androzani" remains one of my favorite episodes of Doctor Who to date.
The most interesting thing we saw when the Doctor made his return was the change in his nature... before he was the ancient traveller looking for things and friends to keep him busy while plunging though the extra-dimensions that we know are there... but he can see and feel. With his return he was an old soldier... lost and alone... the last survivor of a race that no longer is, and struggling to find something to hang on to.
The Doctor's first seasons (at least the new ones from 2005-2009) are now easily accessible via Netflix, and have some fantastic storytelling, almost all of which is self contained in each episode though Russel T. Davies "the Doctor as Messiah" tends to get a little heavy handed in the episodes that he writes.
Steven Moffat (the Doctor's current showrunner) has some of his BEST work in these series. Including Blink, a exceptional piece of television that only actually involves the Doctor tangentially. There are many things to expect from a show like Doctor Who... but the one that I enjoy the most is the un-adulterated creativity that pours into each of them. While I love the classics of Doctor Who dearly... I there is nothing that I can recommend more than the new series. It's fun, scary, brilliant, and shining and while it's regular cast is small even the guests for each individual episode work their utmost to make it an excellent viewing experience for everyone.