How to Introduce Your Friends to Doctor Who (in 4 Episodes or Less)

“Rose,” Series One, Episode One (revived series) 
At first glance, “Rose” seems like the logical place to introduce a new viewer to Doctor Who. Start at the beginning, right? It’s unfortunate, then, that the beginning hasn’t aged very well: sentient plastic, an idea that maybe (???) seemed cool 11 years ago, looks absolutely ridiculous today, and after carefully observing friends as they watch it for the first time, I have the feeling that the pacing seems a little slow too. That’s a shame, because there’s a lot that’s good about the episode—the idea of getting to know Rose first is a great one, and allows the viewer to really experience the full wonder of the Doctor’s world (the next episode, “The End of the World,” does this really well too, and also includes the best use of Britney Spears’s “Toxic” that I’ve ever heard).

“The Christmas Invasion,” Series Two, Christmas Special
“The Christmas Invasion” is David Tennant’s first appearance as the Doctor, and given that he’s one of the most popular Doctors of all time, he’s a pretty good bet for an introductory episode. The problem here is three-fold: first, even though it’s Tennant’s first episode, it relies a lot on Rose’s world, so there’s still a lot of explanation that has to happen for the first time viewer; second, it’s longer than a normal episode, which makes it a little riskier; third, like “Rose,” it hasn’t aged very well. Also like “Rose,” it has a lot to recommend it, but it’s better to wait until your friend is hooked to show it to them.

“The Eleventh Hour,” Series Five, Episode One 
“The Eleventh Hour” was my go-to introductory episode for quite a while, and for good reason: it features a new Doctor, a new companion, and a new showrunner—Steven Moffat. It also brings a cinematic lushness that feels comfortable to today’s television watchers, includes a lot of humor (that fish sticks and custard scene), and doesn’t require a ton of explaining right off the bat. I still think this is an okay place to begin, but the problem is that you run into both the Daleks and the Weeping Angels only a couple episodes later, which means that if you have any heart at all, you have to at least go back in time to “Blink,” and then River also shows up, which means a trip back to “Silence in the Library” and “The Forest of the Dead,” and now there’s a good possibility that both you and your friend are hopelessly confused and that the momentum has been lost.
A better plan
So by now it’s clear what we need—a snappy, funny introductory episode that explains the Doctor, and which has aged reasonably well, but that occurs far enough back in the continuity that you don’t have to jump around in time to show them the episodes that later seasons rest on. Fortunately, just such an episode exists. I present to you:

“The Runaway Bride,” Series Three, Christmas Special 
“The Runaway Bride” is Catherine Tate’s first appearance as Donna (she doesn’t return as a regular until the next series), and it’s charming: mostly fizzy and funny, but containing unexpected dark spots that allow the Tenth Doctor’s delightfully unbalanced homicidal streak to really shine. The chemistry between Tate and Tennant is, as always, brilliant (and if you haven’t seen them in 2011’s Much Ado About Nothing you should rent the recording immediately), and if the effects are occasionally hilariously bad, the quality of the rest of the episode manages to negate them. Most importantly, Donna’s stubborn ignorance means that the Doctor is constantly having to explain himself, so hardly any extra exposition is required.
Now, if you want to, you can simply work forward from there, putting you and your friend directly into Martha’s season. But if you’re looking for something more expedited, a mere three episodes can get you to series five, and it’s all in chronological order:
“Blink,” Series Three, Episode Ten
“Blink” is worth stopping for, even if you’re trying to catch your friend up in a hurry. First, it’s a fantastic episode, frequently counted as one of the greatest Doctor Who stories ever filmed (in a BBC America poll taken around the time of the 5oth anniversary, it ranked number one), and if your friend was at all on the fence about the series, it should settle the matter for good. Second, because the Doctor is barely present, very little explanation is needed—it’s basically a stand-alone episode. Third, it’s the first time we meet the Weeping Angels (ideally, it would have also been the last), and since they’re both quite difficult to explain and quite important later in the series, it’s best if your friend can be there at the beginning.
From here, I would be tempted to proceed to the beginning of Donna’s season, “Partners in Crime,” and simply watch chronologically from there—it’s a solid season with a number of stand-out episodes, and Donna’s exit is so heartbreaking (and relevant to Clara’s exit several years later) that I think it warrants a watch. But if we’re still in a hurry, only two episodes are truly necessary:

“Silence in the Library”/ “Forest of the Dead,” Series Four, Episodes Seven and Eight
This excellent two-parter is a much-watch for one major reason: River Song. Like the Weeping Angels, River is both complicated and important to the series, so it’s best to simply meet her at the start. She’s also a deeply enjoyable character, albeit one who spent a lot of years in an unsatisfying limbo, so she’s a good choice for a beginning viewer.
Now your friend is caught up to the beginning of series five, which seems to still play well with today’s viewers (there’s a really marked difference in visual quality between series four and five, which can make a pretty big difference to people just coming to the show), and can go forward on their own from there. Hopefully, now that they’re hooked, they’ll also go back and enjoy all the wonderful episodes of Russell T. Davies’ tenure—either way, I defy you to avoid rewatching them yourself.