I would say either start from the beginning of the new series, or if you want to plunge into the classic show, look for some "best bits" rather than any kind of systematic approach. If you like that, there's plenty of time to hunt down every obscure story in the classic show's 700 ep history.

Here's a small number of logical starting points:
1) The War Games. This is the last story of the 2nd Doctor, and the one that finally reveals the Doctor's backstory. It's possibly the best of the 60s B&W era. Warning: this one is 10 episodes long, so be prepared for a... leisurely... approach to pacing.
I'm going to recommend several stories from the Tom Baker era, who was the most popular of the classic Doctors, so is a good starting point:
2) Genesis of the Daleks. Not as flawless as some would claim, but still a very good story revealing how the Daleks were created, and introducing for the first time... duh..,. the Daleks' creator, Davros.
3) Pyramid of Mars. Nothing special about this one from a continuity perspective, but it's arguably the best story from arguably the best season the show ever had.
4) City of Death. This story by Douglas Adams is well-loved, despite this season as a whole being very controversial. Unlike some other entries this season, there's a solid, entertaining story along with the clever dialog and humor. This story is the equivalent of sitting down with a glass of wine and having a conversation with an old friend.
And two from the later years:
5) The Five Doctors. This story is, frankly, a mess. But it still has a following, because it was the 20th anniversary special, featured all five Doctors (although they recast the dead one), a ton of guest stars and a who's who of monsters. If that kind of "everything and the kitchen sink" spectacular appeals to you, it may be worth checking out.
6) Remembrance of the Daleks
6a) Curse of Fenric
I have a lot of affection for the 7th Doctor, who was extremely controversial. Either of these stories showcase what his fans love about him, at his dark, manipulative, yet fundamentally decent self.