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Doctor Who is a British TV sci-fi series about a human looking alien ("The Doctor") who travels through time and space with various companions (usually human, young, and female). This setup allows writers to set stories in any setting: historical, futuristic, current. I grew up with it as a child in the 1970s, as did many talented writers who have rebooted it this century resulting in some of the best TV of recent years.
There's much that's been written about it, I've put this page together as my personal overview of the series, in particular where to start if you'd like to explore it without watching every episode.
Doctor Who is one of the longest running TV series, it began in November 1963. You can divide its history so far into two broad periods: Classic Who which runs from its beginning until 1989 when the series was cancelled. New Who runs from its reboot in 2005 until today, where it's still going strong.
If you want to explore Doctor Who from scratch, I would start with New Who. Classic Who has its charms, and some remarkably good serials. But the quality of Classic Who varies from very good to truly awful, and comes with special effects that vary from cheesy to truly awful. So although I'll suggest some Classic Who serials later on, most new viewers should start at New Who. Although there is some fan-service to Classic Who, you can appreciate New Who without knowing anything about Classic Who.
Doctor Who is originally conceived as a children's show, but many adults enjoyed it too. New Who is expressly written as a show that you can enjoy both as a child (occasionally hiding behind the sofa) and as an adult. I started watching Who at 6, and have run into plenty of children who enjoyed it at a similar age.
As I write this New Who has finished its eighth series. To get into it you have two choices: start from the beginning of New Who with Rose (S01E01) and take a completist approach through every episode, or cherry pick the best. There are certainly rewards for the completist approach, with some good internal references and many great episodes that you can't appreciate by cherry picking. The completist approach does require you to invest in a lot of time (eight episodes of 13 episodes each, plus various specials). Also there are some duds in there, although they never get as bad as the lesser Classic Who episodes. Even if you cherry pick, you can pick individual series to watch completely, which is satisfying when there is a good series arc to enjoy. Often the season finales are very good, and can't really be appreciated without seeing at least the one series that they finish.
You don't need any advice to be a completist, so most of this page is for cherry pickers. My basis for picking them out is episodes that are both the best of the series, but also those that don't rely on a series arc to enjoy.
But before I begin, I'll start with a couple of notes for completists. The most important thing to note for completists is not to give up too early. The 3rd and 4th episodes of the first series are two of weakest episodes of all of New Who, and most fans dislike them intensely for their childish humour (involving farting aliens). Stay at least until S01E05 - Dalek - which raises the game considerably.
The first question is where does a cherry picker begin, and I'd always advise starting with Blink, which is on most people's short list for greatest Who episode ever. It actually doesn't feature the Doctor that much, and mostly ignores the companion of that series. But it's a clever plot, superbly acted by a young Carey Mulligan, and lots of wit in the script. Deservedly it won a BAFTA drama award, a rare event for a sci-fi story.  The episode was written by Stevan Moffat, who since went on to be the showrunner for Doctor Who. He's also the co-showrunner for Sherlock, the writer of Jekyll, and I rate him with Joss Whedon as one of the best writers of our time.
So start with Blink, but after that you can mostly pick and mix as you like. I'm going to list my picks with the series chronologically, but you don't have to do them in that order. If any episodes require you to see some others first, I'll point that out. You should get used to different actors playing the Doctor - there is a clever techno-babble reason why multiple actors can play the same character, which doesn't affect a cherry picking watcher. Each actor emphasizes different aspects of the same character. Companions are different people, but again the cherry picking choices I've made don't rely too much on their ongoing story.
The first series was the reboot of Doctor Who, making it return to the screen after a silence of fifteen years. The driving force for the reboot was Russel T Davies, who already had garnered a fine reputation as a TV writer. Playing the Doctor in this series is Christopher Ecclestone, sadly in his only series as the Doctor, with Rose (Billie Piper) as his companion. The standout episode of the first series is Moffat's The Empty Child / The Doctor Dances. Most movies aren't this good, and this double episode introduced the Moffat approach of combining fright and wit. The other cherry pick I'd make from the first series is Dalek, which reintroduces the Doctor's iconic enemy in a story for them that still hasn't been surpassed.
The first series also has one of the better story arcs, so may be worth doing the full series, if only to really enjoy the final two-parter which isn't worth watching without that context. (A tip if you do watch the whole series: don't watch the trailer for Bad Wolf (it appears at the end of Boom Town) as it gives away an important part of the plot of Bad Wolf.)
The second series has David Tennant playing the Doctor. My cherry pick here is again the Moffat episode: The Girl in the Fireplace. I also think the two parter The Impossible Planet / The Satan Pit is worth watching.
The third series continued with David Tenant but brought in a new companion: Martha. Blink comes from this series, but another outstanding highlight of series three is Human Nature / The Family of Blood. You do need to have watched a few Whos to really get into the Doctor's nature and character to appreciate this one, (I'd suggest watching the series 4 picks first). It was the first episode for me to reach the same heights as The Empty Child
For the fourth series Tenant was joined by the already well-known comedienne Catharine Tate as the companion Donna. Again Moffat came up with an outstanding thriller/comedy Silence in the Library / Forest of the Dead. But there are also other great highlights here. The Unicorn and the Wasp is probably the most out-and-out comedy in Who, a wonderful send-up of the Agatha Christie country house murder mystery (feature Agatha herself, as only Who can). Russel Davis also came up with a taut character-driven thriller Midnight. I also really enjoyed Turn Left, but am not sure whether to recommend it here as it makes many references to non-cherry-picked episodes. However I think you can still enjoy it without really following those references, if only for some great acting from Tate and Bernard Cribbins.
After the fourth season there was a year of specials, which generally isn't counted as a series. From this set, I'd pick The Waters of Mars, which was a fine take on the "base under seige" style of Who plot.
At that point there was a wholesale change. Russel Davies gave up the role of showrunner, handing over to Moffat. Tenant also gave up the role of the Doctor. Matt Smith became the eleventh Doctor, and we got a new companion in Amy, later joined by her husband Rory. The first two episodes of the Moffat era [TODO: names] are both worth picking. I also would pick the two parter [TODO: name], although for this one it's important to have seen Blink first (to know about The Weeping Angels) and Silence in the Library / Forest of the Dead to know about River Song. River Song becomes an important character in various episodes in series 6 and 7 after this. Series 5 is also another good series to watch all the way through, with good development up to the finale. But if you want to stay with cherry picking, do watch Amy's Choice, and perhaps Vincent and the Doctor .
One of the traditions of New Who is to have a Christmas special episode for broadcast on Christmas Day. Davis did the earlier christmas specials, and I don't put any of them on my pick list. But I do pick Moffat's first special [TODO: name], as a clever remix of the Scrooge story starring Michael Gambon.
With series 6, Moffat decided to break the mould and start the series with a two parter The Impossible Astronaut / [TODO: name], which opens the series with a bang. The unresolved question from this episode may entice to watch this whole series, and that's not a bad choice. If you don't, you shouldn't miss The Doctor's Wife (written by Neil Gaiman) and The Girl Who Waited.
Series 7 is a series of two distinct halves, with a halfway split seeing the departure of Amy and Rory and a new companion Clara. My cherry picking suggestion here is to see The Asylum of the Daleks first (which sort-of introduces Clara). Then see The Snowmen which picks up the Doctor after he is sadly separated from Amy and Rory and sort-of introduces him to Clara. Moffat does some clever meta-textual stuff here, which you can only appreciate fully if you're a completist but those two episodes still stand strong even without fully getting the meta-text in the background. I would also pick out The Crimson Horror, which is a fun comedy that features a great performance form Diana Rigg.
Series 7 finishes with the Big Event episode of Who so far, the 50th anniversary special, broadcast 50 years after the first episode. Naturally there's a huge amount of references for the fans in this episode, which unites both Tenant and Smith's Doctor together with John Hurt's sort-of Doctor. It still stands alone for pickers, so don't let the lack of background stop you watching it. Before you do, however, catch the Minisode: the Night of the Doctor on youtube. It's a remarkable coup of story-writing to get so much into a six minute episode (although it does help to know that Paul McGann was the 8th Doctor, the one that immediately preceded the New Who period).
With series 8, [TODO: name] takes on the role of the Doctor, and brings a darker, more grumpy Doctor to the scene. Not everyone liked this take on the character, but I did as it reminded me of My Doctor (the 3rd - Jon Pertwee). It also is my favorite full-series of arc, with some great character development and interplay between the Doctor and Clara. For cherry pickers, however, I'd pick out Listen, [TODO: name], Into the Dalek, and [TODO: name], but ensure you see the others before you see [TODO: name] since that last episode gains a lot from getting a sense of the nature of this new take on the Doctor.
As I write this, the last Doctor Who episode was the last Christmas Special, which is also a pick for its wonderful mashup of Alien, Santa Claus, and Inception.
So how about Classic Who? There's lots of Classic Who, but if you're going to explore it, I should pick out a couple of places to start. Unlike New Who which goes for single or double episodes, Classic Who had short serials of 4-6 half hour episodes. I would start with City of Death, which has the iconic Tom Baker as the fourth Doctor, Romana as a Time Lord companion, and a script that clearly shows it was part-written by Douglas Adams. After that I have to point you to some 3rd Doctor (since he's my doctor), and I'd pick out Carnival of Monsters, written by Robert Holmes, generally rated as the greatest of the Classic Who writers, and features Jo Grant who was the companion I best remember.
1: Blink also achieves Doctor Who's higher purpose. You see Doctor Who is only secondarily about entertainment, it's primary purpose was always to scare the living daylights out of small children. I may be too old now to get behind the sofa, but I do remember how much I enjoyed it.
2: Vincent isn't one of my favorites, but many Whovians like it. It has a different appeal to most of my other picks, so is worth it if only for a change of pace.