If you're just starting brand new to the series, it's probably best to start with the new series. (I'm a huge fan of the classic series but it can be very much an acquired taste. It's got the pacing of the old Dark Shadows
soap opera and production values that make classic Star Trek
look like Avatar
by comparison. But the characters are very interesting and it's a very unique style of storytelling.) If you feel a larger interest in the franchise as a whole and have an inordinate amount of patience, you'll find classic Doctor Who
very rewarding. But if you're just starting out, you might want to start with the newest stuff.
As for where to start with the new series, opinions will differ but I think the current Doctor, Matt Smith, is the best of the 3 new Doctors. Not only is Smith the most interesting and eccentric of all the Doctors but I think he's had some of the most interesting & charismatic companions: Amy Pond, Rory Williams, River Song, & Clara Oswald. And, I think the Smith episodes have done the best job of incorporating time travel into the stories themselves, (whereas the show normally just uses time travel to drop our heroes into exotic locations week after week). So, taken that way, my favorite timey-wimey episodes of the Matt Smith years:
"The Eleventh Hour"
(The 1st appearance of Matt Smith, Amy, & Rory.)
"The Pandorica Opens"/"The Big Bang"
"The Impossible Astronaut"/"Day of the Moon"
"The Girl Who Waited"
Also, a couple other favorites of mine from the Smith era:
"Vincent & the Doctor"
(A surprisingly touching story where the Doctor & Amy meet Vincent Van Gogh. I always tear up a little bit towards the end.)
(By far the funniest of all the Smith episodes, displaying the 11th Doctor at his most eccentric.)
"Nightmare in Silver"
(The Doctor gets partially posessed by an advanced Cyber-intelligence. Matt Smith gives his best performance ever, contrasting the Doctor with the evil Cyber-intelligence fighting for control of his body.)
I could go on all day just talking about how much I love the 11th Doctor, but there's lots more about the series to enjoy.
David Tennant often comes accross as the most popular of all the Doctors. (There was a favorite Doctor poll in Entertainment Weekly last year where Tennant absolutely destroyed the competition.) I like Tennant a lot but he's sometimes a bit over-the-top for my tastes. Also, I was quite annoyed by all the romantic mooning that his companions tended to do over him. But, if you're a girl, you'll probably think he's cute and he'll instantly worm his way into your heart. IMO, Tennant's run was far more uneven than Smith's but there are some definate highlights to his era:
(Reuniting the Doctor with Sarah Jane Smith, one of his old companions from the 1970s.)
"The Girl in the Fireplace"
(A time travel story very much in the style of some of the better Matt Smith episodes.)
"Human Nature"/"The Family of Blood"
(A heartfelt story where the Doctor erases his memory and lives life as a human.)
(Tennant's most popular episode, by far, despite the fact that the Doctor is barely in it. This is the episode you show to people even if they don't want to watch the rest of the series.)
"The Unicorn & the Wasp"
(While Tennant's other companions, Rose & Martha, tended to get on my nerves, Donna was a great bubble of unpredictability. Just watch the poisoning scene.
(A small, claustrophobic story where the Doctor & a group of strangers are trapped aboard a tour bus while menaced by a non-corporeal alien.)
"The Stolen Earth"/"Journey's End"
(If you love crossovers, man, you'll love these episodes. All the recent companions come back, plus people from the spinoffs!)