The Christopher Eccleston (9th Doctor) season is the best place. It doesn't get good for a few episodes but episode 9/10 are an amazing two-parter which you could honestly just try out standalone if you know the basic concept of Dr Who. Even as a big fan of Dr Who, I used to get up at 6am on a Sunday to watch the repeats of the old serials with my Dad then go back to bed, I would suggest watching a couple episodes of the 9th Doctor (unless he in particular appeals to you) then go on to the 10th Doctor (David Tennant) which is a much more consistent show since they have had time to perfect and practice it. So I would suggest you watch:
Ep. 1 - Rose
Ep. 6 - Dalek
Ep.9+10 - The Empty Child/The Doctor Dances (The best episode in the Series)
Ep.12+13 - Bad Wolf/Parting of the Ways (The Series finale)
Series 1 Christmas Special (Introduces David Tennant).
Then go onto the Tenth Doctor and just watch the episodes in order. If you get really into it then go back to the first season of the revived series which is enjoyable.
When you start to really get into Dr.Who I would still not suggest just watching from the very, very beginning since its a different show and would instead suggest you start with the best episodes and when you find an older Doctor you really like just watch their run through. There is almost no continuity in the show early on so its not really a problem. I would suggest starting with Tom Baker and specifically City of Death or Genesis of the Daleks.