(Stay with me: there's a point to this story) The first time I had sushi I was a guest in some sort of luxurious retreat in Japan. I think each meal cost some $200-300 per person (in grad school I ate Jack in the Box for a whole year, to give you a sense of my wonderment). Needless to say, the meal was totally, utterly wasted on me. I actually disliked it, in fact. Nowadays I love sushi, yet to this day I can't pinpoint when I had that second meal of sushi. I just remember always loving sushi, except for that first meal. Somewhere between that first and the second meal the revelation came, unforced and unannounced.
Dr. Who was the same way for me. My first episode was David Tennant's second (I love love LOVE Tennant btw) and after it was over I just sat there scratching my head in bemusement. I was with my family-in-law so you know, when in Rome .... I thought it was a bit naff, to be honest (Earth is saved thanks to ... I won't give it away, but come on: a beverage?!). Now I absolutely adore Dr. Who, yet I can't remember when I watched that second episode. I just seemed to always love it, except for the first time.
So my advice is: pick the first episodes of any particular actor who plays a modern Doctor -- i.e. Christopher Eccleston, David Tennant, Matt Smith -- and if you don't like it outright, set the other episodes aside for awhile. Come back to it when you have a particularly boring afternoon when you can't think of anything interesting to do, and then watch the next episode. You'll probably find you have come to love it already!
P.S. I happen to like Tennant and Smith, so those get my vote for a first Doctor. Actually I think Smith may be the best of the modern Doctors, but this is so highly dependent on the quality of the writing that it's safe to say that if you don't like one of the three, you won't like the other two.
P.P.S. And, if you missed it: as others have mentioned, if you're an Amazon Premier member, all the modern Doctor Whos are free to watch on Instant Video.