Doctor Who, on BBC One (UK) / Sci-Fi (USA)
Caution: spoilers!
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The original Doctor Who TV series ran from 1963 to 1989. The new series began in 2005, in the U.K., but I didn't see it until 2006, when it aired in the U.S., on Sci-Fi Channel. (That network would air the first four seasons, before American rights were picked up by BBC America, which had already been airing reruns of seasons 1-4, and then became the primary American distributor of the series from season five onward.) So... Quick recap of the old series: there's this human-looking alien called "the Doctor," who is a Time Lord from the planet Gallifrey. He has a time machine/spaceship called the TARDIS (Time and Relative Dimension In Space). The tardis is disguised as a 1950s British police box (which is like a phone booth), but it's bigger on the inside. He uses it to travel throughout time and space, and has lots of strange and exciting adventures, sometimes even on Earth. Sometimes he travels alone, and sometimes with one or more Companions. The Doctor is centuries old, and occasionally regenerates a new body, so he can be played by different actors. But it's not just his appearance that changes, he also gets a new personality when he regenerates... but the one constant is that he's always a hero (and kind of a rebel). But Time Lords can only regenerate twelve times (under normal circumstances), for a total of thirteen lives. And this new series begins with the ninth Doctor. (Well, we'll call him the ninth Doctor, anyway. He might technically be the tenth, but that's a story for another time. Same rule applies to all subsequent regenerations in the current revival of the franchise.)
Anyway, when I first started watching the series, I didn't like it as much as my recollection of the old series, which I'd grown up watching in the 1980s, much of it in reruns (Though there are plenty of older episodes that I've still never seen, I've seen at least a little bit of every Doctor, and a great deal of most of them.) I did think the production values of the new series were much better than those of the old, and I did like the new show well enough, but I also thought it could sometimes have a bit of the familiar cheesiness of the old show. But also like the original, there are some great stories and characters and it's just really fun (and sometimes a bit scary). And the new show grew on me over time. I'd definitely say it gets better as it progresses. One great thing about this series is the level of continuity. Like, there are major things referenced throughout a season or seasons, which may be totally obviously leading up to something, but then there are also things established that you may not even notice or think about beyond their importance to what's going on in the episode (or things that seem to be of no importance whatsoever), and later those things may come together in unexpected ways, leading up to something quite major. Bloody brilliant. (This is something I may not have really been aware of until the end of season one, but once the trend was established, it was easier to notice it in subsequent seasons... or at least it's more expected.) And in fact, when I started rewatching season one, ten years after it first aired, I thought it was much better than I remembered. Seriously, there is an amazing level of poignancy to it.
Ninth Doctor (Christopher Eccleston)
Series One (2005)
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The Doctor gets a new Companion, a 19-year-old shopgirl named Rose Tyler (Billie Piper), who comes from 2005 London. When she joins him, they have adventures throughout time and space, though they do occasionally stop back into Rose's time and place, where they also end up having adventures. And they visit Rose's mother, Jackie, and also Rose's friend/ex-boyfriend, Mickey Smith. (Rose's father, Pete, died when she was a baby, but we'll see him in one very emotional episode.) Anyway, throughout this season (and the next), Rose turns out to be one of the best Companions the Doctor has ever had. She's basically an ordinary 21st century girl, but she has the same sense of adventure as the Doctor, she's rather clever and daring and can take charge of situations when she has to. Once she's gotten the hang of things, she quickly becomes accustomed to times and places and beings that are beyond her imagining. She's definitely a match for the Doctor, a good friend, and... you know, unlike some of the Companions he's had in the past, she's quite helpful. You know, she serves a purpose beyond just being someone for Earthbound viewers to relate to, in the midst of all the madness of the Doctor's life. (Which is not to say he hasn't had some great Companions before, because he definitely has.) Another interesting thing about the new series is that it occasionally explores the consequences of the Doctor's lifestyle on the friends and family who are left behind when someone becomes his traveling companion.
I should also say that a lot happened between the end of the original show and the start of the new one (or actually, between this and the 1996 TV movie that came after the original show). At some point, there was a huge Time War between the Time Lords and the Daleks, and both races were pretty much completely annihilated. (I imagine that people who read all the novels, listened to audio dramas, and so forth, would know more about it than I, since I haven't read/listened to any of that. But maybe such things don't actually say much about the Time War. I don't know.) Anyway, the Doctor is the last of his kind, which imbues the new series with much of the poignancy that I alluded to, earlier. Of course, most of the time the show is just fun and exciting and there's a lot of humor. But references to... well, the Doctor's loneliness and regret... pop up from time to time. Meanwhile, lots of other stuff happens. One big thing is that the Daleks (who were supposedly extinct) appear a couple of times in season 1. And at one point (in the very distant future) we're introduced to an ancient being that appears to be just a giant head, who is called "the Face of Boe." (We'll see him again in both the second and third seasons.) And... I'll refrain from mentioning too many plot developments, but I must mention that the Doctor and Rose are joined in their travels, for a few episodes, by a former Time Agent (now a con man) from Earth's 51st century, called Captain Jack Harkness. He's left behind at the end of the first season, but we'll see him again, eventually. I also need to mention that throughout the season, there were a lot of references to something called "Bad Wolf," the meaning of which is finally made clear at the end of the season. Also at the end of the season, the Doctor regenerates, so the second season will feature the tenth Doctor.