Start with Eccleston in 2005. I didn’t mind him, and I think that series is pretty strong. I really didn’t like Tennant, though; I see him as the Edward Cullen of Doctors. I enjoy Smith immensely!
But you may enjoy Tennant and it’s quite legitimate to do so 
After them, if you’re interested in the old stuff, start either from the very beginning with Hartnell and Troughton or go to Tom Baker. Baker’s Doctor is very accessible, but I utterly adore Hartnell and Troughton.
Bowties are cool.
And miss “Dalek”??
I’d agree with some of the othersa and say just go with the start of the Eccleston run and go from there. Adding that even in seasons that you might not enjoy as much overall, there are some really standout episodes in all of them (particularly those written by Steven Moffat, which should get you looking forward to the Smith seasons where he takes over as showrunner).
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Start with Eccleston. The Doctor changes in each incarnation, but what came before is important. The dark, broody Eccleston who learns to lighten up with Rose beside him. Younger Tennant has regained some of the joy the Doctor feels with humanity -
I always felt the Eccleston Doctor was saving humanity because that’s what he’s always done. The Tennant Doctor saves humanity because he really loves humans, even as he sees our faults, he also sees our greatness. The Smith Doctor really becomes the Emelia Pond Show, IMHO. I like Pond, and I love Rory, but taking over the whole intro to the series? Too much.
Anyway, start with Eccleston.
StG
I will not reveal my feelings for Amy Pond. Its a private matter between me and her.
Start with “Rose”, the first Eccleston episode in 2005 and just work your way forward. It was designed as a reboot of the series for people who knew nothing about Doctor Who, and that first season introduces a lot of characters and enemies that will be important later. If you skip ahead, you’ll miss stuff and be confused.
And there’s nothing wrong with Eccleston. He’s lots of fun, and does a great job with the character.
That gives you seven seasons of new material to watch. And if you still want more, then you can go back to watch classic Who. I’d suggest jumping back to the Tom Baker (he’s the one with the scarf), and working your way forwards and backward. That was when the series was at the height of its popularity.
Oh, and whatever you do, DON’T BLINK. Blink and you’re dead. (You’ll understand what I mean soon enough. :D)
Start with Rose, the first Eccleston episode.
Also, this reminds me, are there any Doctor Who fans who’ve also seen The Mentalist? I keep thinking Matt Smith’s Doctor seems inspired by Patrick Jane.
But…Tom Baker has a scarf!!
Definitely start with Ecclestone in “Rose” - the whole point of that season was to reboot the show and serve as an introduction for new viewers, so it’s the best place to begin.
It helps that Ecclestone’s my Doctor.
Is this about the voiceover at the start of each episode? If so, that was on the BBC America ones only. The native BBC showings didn’t have this, so try and watch those.
Tom Baker is still the best, but if you didn’t see that really as a kid, you’re much better off starting with the new 2005 season and going from there.
Scarves are also cool.
My wife just discovered Doctor Who last fall. First we watched all the newest episodes, but then we went back and started watching the original run. The Tom Baker episodes are amazing, but you’ve got to be willing to look past the shoestring production values and the “nail it in one take” attitude of the BBC back then. They’re absolutely worth it, but I wouldn’t recommend them to someone trying to get into the series for the first time.
I have a knitted 16-foot Who scarf
Scarves are cool, but <bow>ties suck, at least Eccelston never wore a hangmans noose during his stint as Nine…
Oh, and the Baker-era episode “genesis of the Daleks” still holds up well, with the dalek’s last statement…
“we are entombed, but we live on, we will prevail, we wil grow stronger, when the time is right, we will emerge and take our rightful place as the supreme power of the univerrrrrssseeee!”
Start with Eccelston. That’s the only way that you’ll guarantee that you won’t miss the farting aliens.
Not true. Sara Jane meets 'em, too. 
Meh. Too whingey by half.
Plus, DW should never do Americans.
Mine is only ten feet or so but I loves it! 
Oh, and to the OP, yes, start with the 2005 reboot and go from there.
In case this hasn’t been settled for you yet, here’s my take.
Got into Who just last year and a couple months. Started with 2005 and watched them all back to back to back on netflix, up through season 4. Then watched the Matt Smith season 5 episodes, and then 6 as it came out on TV.
The first season, with Eccleston, is certainly not the best but I would highly recommend not skipping it. David Tennant’s seasons are the absolute best. Matt Smith is awesome too but you’ll be missing out so much phenomenal TV if you don’t watch the Tennant episodes that I will cry for you over it. Watch 'em all back to back, as many episodes as you can manage to cram in a day. That’s the best way to watch Who (or any TV show I guess).
Then go back and watch some Tom Baker episodes. I’ve seen a few of his arcs and they are enjoyable and fun and interesting.
I tried watching some of the first doctor but I couldn’t take the acting.
But seriously. Doctor Who is some of the best television you’ll ever see in your life. Not every single episode of course, but overall, I came away feeling like I had experienced something uniquely phenomenal when I got through the first four seasons on Netflix.