It's a question involving several variables: how much time are you willing to commit to catching up? How tolerant are you of old TV shows with more talking than action? Are you able to look past a low budget and able to enjoy what the production was trying to achieve?
In a perfect world, I'd expect your commitment to the project to be 100% and advise you to start with "An Unearthly Child" -- the very first episode from 1963 and follow the series forward linearly as concepts developed. It's amazing how much has been retained from the early days.
But if you're like me, you're not going to enjoy doing it that way. Look, I like the show, but there are times when the low production values are so glaring in the old days that you can't help but laugh at the supposedly terrifying monster. Heck, even as a teen back in the seventies when I first started watching on PBS, I realized some British programming was ... an acquired taste.
Still with me? Terrific! Start with the 1996 Paul McGann movie. Yeah, I know, it's the failed pilot for a new Fox series that never happened, and yeah, it was made by Americans, but it really set the tone for the future of Doctor Who beyond that point. For the first time, the franchise was given a decent budget and special effects, and it introduces you to basic concepts of the show. You even get to meet one of the earlier Doctors.
After that, watch the first year with Chris Eccleston as the Doctor and then "The Christmas Invasion." The latter introduces David Tennant in the lead role. Then watch the first two episodes of Tennant's first year, "New Earth" and "Tooth and Claw".
Stop there. By now you should have a decent feel for the show and if you really, really like it, you need to get a little wibbly-wobbly with your viewing order and take your first dip into the show's distant past. Find "The Time Warrior", a Jon Pertwee episode from 1973. Then "Planet of the Spiders", "The Pyramids of Mars", and finally, "The Hand of Fear". However, skip all that if you've decided the show isn't quite your cup of tea.
Now watch "School Reunion" through "The Runaway Bride". You should notice the show's quality steadily improving, and David Tennant is generally one of fandom's favorite Doctors. By now, you should get the show even if you don't like it. If not, you're thick and there's no hope for you, but that's not relevant here.
Now move on to "Smith and Jones". Martha Jones is an unappreciated companion, but I love it when the Doctor has someone along who doesn't need to be talked down to. Plus, this is a fantastic season/series of Doctor Who with some really stand out episodes, including "Family of Blood" and "Blink". Finish off the season, then find "Time Crash". If you're curious about Peter Davison's Doctor, go wibbly-wobbly again (you should get that reference after having watched all the above -- and if you don't, you're thick) and go find Tom Baker's last regular performance as the Doctor with "Logopolis", then follow up with "Castrovalva" through "The Caves of Androzani" and don't forget "The Five Doctors" along the way. Stop before you get to Collin Baker ... he's a bit of an acquired taste of acquired tastes, and while I really like some of his episodes, I loathe his companion and wouldn't wish her upon anyone.
Anyway, now pick up with "The Voyage of the Damned" then "Partners in Crime" this re-introduces my personal favorite Doctor Who companion and I really enjoyed most of the season/series that followed. Make sure you catch "Silence in the Library" and "Forest of the Dead". Seriously, those are important episodes. And finish off the season with "The Stolen Earth" and "Journey's End". If you haven't noticed that the final episode of each series/season tends to get really high-concept and weird by this point, you're thick.
A series of "specials" follows with "The Next Doctor", "Planet of the Dead", "The Waters of Mars", and "The End of Time", which is the end of David Tennant's stint.
Are you into the wibbly-wobbly, timey-whimey stuff? Then go back and watch Patrick Troughton's "The War Games". It's one of my favorite episodes. You might also enjoy "The Two Doctors".
Now you're ready for Matt Smith in "The Eleventh Hour". It is my opinion that Matt Smith is the best Doctor yet. I'm not alone in that opinion, but neither is that thought universally shared. Still, Smith's first season/series is a lot of fun and a bit tighter than most of the ones that preceded it.
By now, you're a Doctor Who fan, or you're not, or you're really, really thick. In any event, I hope you've enjoyed the trip. Because that's what it's all about.
One more thing: while Doctor Who is loads of fun, don't go in expecting a serious show. While it isn't comedy, it doesn't take itself too seriously, either. This isn't hard science fiction, it's a family-oriented, light-hearted affair with some really amazing ideas, and is sometimes the best science fiction to ever air on television.