|A picture is worth a thousand words.|
There are two ways to start. You can start with the classic series, S1 E1 Doctor1. Or you can start with the new series, S1 E1 Doctor9. The classic series is, well... classic. The new series is the only one I've watched, and can offer knowledge about. So we'll go with that.
Doctor Who is a very deep and wide Fandom. That can make it intimidating to start watching. So the best eps to start with are self-contained and have characters who are new to the world of The Doctor.
A perf example is "Blink, S3 E10." It's a one off, and the protagonist Sally Sparrow is new to the world of Who. "Blink" also has one of the scariest baddies in Who history: the weeping angels.
This is the ep I used to introduce my BFF to Who. It took 39 minutes before she became a die hard fangirl. In other words, success.
Another good episode is "The Runaway Bride, S3 E1." Donna is funny, sympathetic, and loveable. If your potential Whovain cares about the characters, they're more likely to want to watch more.
A lot of people skip the Ninth Doctor, and start with Ten. I do not recommend this. And by "do not recommend this," I mean I will come through the computer and smack you with a rolled up newspaper, and say "Bad Fanling! No post for you!" Even if you choose to start with "Blink" or "The Runaway Bride," go back and start with the new series featuring Nine; it's definitely worth it.
When choosing other eps, think about your potential Whovain. "Blink" was a good choice for my BBF because she likes horror. Horror = scary, scary = weeping angels. If your potential Whovain is a romantic comedy fan, something like "Love & Monsters, S2 E10" or "The Unicorn and the Wasp, S4 E7" might be a better choice.
Other than that, S1 E1 is always a good starting point. Just make sure you give a little back story before you do anything.
Gotta dash. Things happening. Well, four things. Well, four things and a lizard.