Children of the Gods is the first episode (or two episodes according to the official numbering, but broadcast as a single, double-length episode, so I'll be referring to the next episode as episode three) and it picks up some time after the movie. Possibly three years.
The Stargate, which has lain dormant since then, suddenly activates and some dudes (including Teal'c) come through, kidnap a woman, and kill a bunch of people before leaving. This prompts General Hammond, now in charge of the Stargate programme, to bring Jack O'Neill, Kawalsky and Ferretti (characters from the movie all now played by different actors, so you've got to wonder why they didn't just form a new team) on board.
Hammond threatens to send a nuke through to gate, forcing Jack tor reveal that Daniel Jackson (also from the movie, also recast) is still alive and living on the other side. At this point they believe that the gate can only facilitate travel between Earth and Abydos, the planet from the movie. All of this setup would probably seem really weird to anyone who hadn't seen the movie.
They decide to send Jack and some through to see Daniel and find out what's up with these bad guys, and he's all like "Cool, but I get to pick my team" and Hammond's all "Nope, you've got to take Sam Carter" and then Carter comes in and - woah, hold on a second, she's a girl! See, they said her name was Sam and everyone assumed that was a boy's name but she's actually a girl! I guess this was mind-blowing in 1997?
Jack's not happy about her being on the team, but don't worry, it's not because she's a woman, it's because she's a scientist and he fucking hates scientists and all they stand for. But it's OK because she's also killed a bunch of people so she might be one of the good ones. This is a really weird scene. So they go through the gate and meet Daniel Jackson and all his friends.
Meanwhile, elsewhere, that woman they (the bad guys) kidnapped back at the start of the episode has been changed into a dress (that is, made to wear a dress not made into a dress) as is dragged before the head bad guy, who has glowy eyes and a weird deep voice. As an aside, he and the other extraterrestrials in this episode speak alien language as they did in the film, but that's going to be forgotten for the sake of convenient very rapidly.
Anyway, the head bad guy (who I may as well start calling Apophis since that's his name and I know it is and this TV show came out 20 years ago) like, shoots some golden light out of his hand at the woman and tells her she "could be the vessel for [his] future queen". She gets all hypnotised an they rip her dress off (prompting one to wonder why they put it on her in the first place) and lie her down.
Back on Abydos, Daniel is showing everyone this cool room he found with all these symbols on it that he says are instructions for using the Stargate to go to other places, and Sam's like "Yeah, that is totally plausible and now I can do some computer magic and the gate can go to other places, not just here." This makes no sense since they supposedly had tried every combination and only one worked, but now there are more combinations, or something? Whatever, new planets every week.
An now Apophis comes to Abydos and starts wrecking everyone's day (but not the main cast 'cause they're in Daniel's special room) and kidnaps Daniel's wife (new actor) and that kid that Jack was friends with in the movie (actually played by the same actor for a change). So Daniel tells the Abydonians to bury the gate for a year and then he'll come back and let them know if it's safe, and he goes with the other Earth-humans back to Earth.
One of the survivors of this latest attack is able to identify the symbols that Team Apophis used to leave Abydos, so Hammond asks his BFF, The President, if he can form some teams to go attack the aliens and The President says OK, so they do that, and they're called SG-1, SG-2, etc and that's why the show's called that. Of course the first team is Jack and Sam and Daniel an they're going through on rescue mission.
Meanwhile they've got Daniel's wife in her underwear in that room where I guess they bring all the women they kidnap, and Apophis says "this one has spirit" so you know he's a creep (if you still weren't sure) and does the hypnosis thing on her. Then they strip her naked and lie her down (Teal'c still watching) and the jewellery woman comes back and does the thing again, only this time the goa'uld approves and crawls all over her and it's gross. Jewellery woman seems pretty into it though. Then they flip Daniel's wife onto her stomach and the goa'uld burrows into her neck and Teal'c shakes his head like he doesn't approve.
Meanwhile, SG-1 finds some people who assume that since they came through the gate they must be gods, and bring them back to their town for dinner. But then Apophis and his crew come in to show off that Daniel's wife is now Apophis's wife, and she's got the glowy eyes now all and all that. SG-1's pretty confused by this turn of events because they don't know goa'ulds are brainslugs yet and they get captured.
They spend a bit of time in jail, where Jack's friend from the movie is and Apophis and Daniel's wife (now Apopis's wife, Amonet) comes by to choose some people to be their children, and of course they pick Jack's friend (among others). Apophis then orders the rest (including SG-1) to be killed.
Writing it out, a lot seems to have happened in this episode, but it was a really long episode so it's actually pretty slow to watch, and the only stuff that really stands out does so for the wrong reasons (ie. unnecessary nudity and weird treatment of women) but as a first episode it does a pretty good job of setting up the premise and letting you know what to expect going forward. Unfortunately, what you can expect is not very good, so if I were watching for the first time I'd be hoping for some rapid improvement. I'd probably keep watching anyway though.