SkunkyArt | Licunatt — Time Loops

Blink (Season 3, Episode 11) – A “Doctor-lite” episode. An entirely standalone story centered around Sally Sparrow (played by a pre-fame Carey Mulligan) and her run-in with one of The Doctor’s most feared enemies.

Rose - Now that you're been introduced to some of the concepts of Doctor who, time to soak in the first episode and a bit of it's camp. The show is rather good but there is still some of that out there and you have to be ready for it.

The Empty Child (Season 1, Episode 9) - Travel to 1941, at the height of the Blitz while homeless children (living on the bombsites) are being terrorized by an unearthly child. Meet Captain Jack Harkness. (Part 1)

The Doctor Dances (Season 1, Episode 10) - The Child’s plague is spreading throughout the wartime London. The Doctor and Rose form an alliance with Captain Jack. (Part 2)

Dalek (Season 1, Episode 6) - Meet the Doctor's mortal enemy & learn a bit about his relationship with them.

The Eleventh Hour (Season 5, Episode 1) - Get a bit of a taste of the style of the (current) head writer of Doctor Who and meet Matt Smith as the Doctor.

These episodes do give you a good feeling about who the doctor is, how cool, lame, scary, cheesy, cheeky the series can be and lets you see the first 3 doctors of the modern series.
David Tennant (series 2-4) isn't very prominent on this list because he's sort of like eating chips/crisps... you can't have just one but season 1 gives you lots of things to be excited about and covers some of the basics. Rose is a good companion to get started with too since she's got a more neutral personality and it's easy to take things in with her. Once you start season 2 and get on the Tennent train, it's a rather fun ride since he's more light-hearted then Eccleston (first Doctor of the modern series) but Eccleston has his great moments too! These are great episodes that skip some of the heavy stuff that you can soak in slowly as you watch the rest in order... if you fancy it.