As others have said you can start from the beginning of the 2005 run. You probably want to watch the new series in order, especially as during Matt Smiths run the story arcs get ridiculously convoluted. (which can be fun if you are into that).
If you are interested in old series, as others have suggested I'd watch the first episode (An Unearthly Child) which is only a half hour. Then skip a couple of episodes to the first Dalek serial (Called either the Mutants/the Daleks). Then watch the Dalek Invasion of Earth, which is generally regarded as a classic.
Then I would watch one serial from each of the seven initial Doctors to get a feel for the style of each one. (There are a ton of "Best Doctor Who Episodes" on the web)
Don't worry about continuity. The producers never did! All you need to know is:
1. The Doctor is an alien, a time lord, with a spaceship (the TARDIS) that travels in time. He is said to have stolen the Tardis, but this happened way before the first episode and the reasons why are not clear.
2. The TARDIS once had the ability to change shape to blend in with whichever time period it arrived in. Since it is now old and this circuit is broken and it looks like an old London police box.
3. The TARDIS frequently goes wrong and end up taking the cast to unusual places.
4. As an alien the Doctor can regenerate into a different body when his current body is damaged (allowing the producers to change actors whenever necessarily)
5. The Doctor will have one or more companions travelling with him. These are usually from present day Earth (though occasionally they've had human-looking aliens, other timelords, or human from the past or future). Generally, though the acting skills on display vary wildly, you don't really need to know anything about these characters back-stories, they're mainly supposed to be ordinary people (If it is important for the episode to know that Sarah-Jane was a reporter, Sarah-Jane will mention she's a reporter early on in the given episode)
6. There was a period (of about three-four series) where the Doctor was exiled to Earth by the timelords. During this period he couldn't use the TARDIS, but still helped save the world from aliens every week. He was working for UNIT, an alien fighting military organization. UNIT often features in modern day stories.
7. The Doctor has several recurring enemies A) THe Daleks look like pepper pots and want to destroy all non-Dalek life in the universe. B) The Cybermen are basically an earlier version of Star Trek's Borg C) The Master is another timelord who wants to rule the universe.
Thats probably enough to understand basically whats happening in any episode. Since the series is for kids anything else you need to know will probably be explained during the episode. (Again ignoring the Matt Smith episode which do require a much deeper knowledge)
If you are interested in old series, as others have suggested I'd watch the first episode (An Unearthly Child) which is only a half hour. Then skip a couple of episodes to the first Dalek serial (Called either the Mutants/the Daleks). Then watch the Dalek Invasion of Earth, which is generally regarded as a classic.
Then I would watch one serial from each of the seven initial Doctors to get a feel for the style of each one. (There are a ton of "Best Doctor Who Episodes" on the web)
Don't worry about continuity. The producers never did! All you need to know is:
1. The Doctor is an alien, a time lord, with a spaceship (the TARDIS) that travels in time. He is said to have stolen the Tardis, but this happened way before the first episode and the reasons why are not clear.
2. The TARDIS once had the ability to change shape to blend in with whichever time period it arrived in. Since it is now old and this circuit is broken and it looks like an old London police box.
3. The TARDIS frequently goes wrong and end up taking the cast to unusual places.
4. As an alien the Doctor can regenerate into a different body when his current body is damaged (allowing the producers to change actors whenever necessarily)
5. The Doctor will have one or more companions travelling with him. These are usually from present day Earth (though occasionally they've had human-looking aliens, other timelords, or human from the past or future). Generally, though the acting skills on display vary wildly, you don't really need to know anything about these characters back-stories, they're mainly supposed to be ordinary people (If it is important for the episode to know that Sarah-Jane was a reporter, Sarah-Jane will mention she's a reporter early on in the given episode)
6. There was a period (of about three-four series) where the Doctor was exiled to Earth by the timelords. During this period he couldn't use the TARDIS, but still helped save the world from aliens every week. He was working for UNIT, an alien fighting military organization. UNIT often features in modern day stories.
7. The Doctor has several recurring enemies A) THe Daleks look like pepper pots and want to destroy all non-Dalek life in the universe. B) The Cybermen are basically an earlier version of Star Trek's Borg C) The Master is another timelord who wants to rule the universe.
Thats probably enough to understand basically whats happening in any episode. Since the series is for kids anything else you need to know will probably be explained during the episode. (Again ignoring the Matt Smith episode which do require a much deeper knowledge)