The premise is that
Khan is...well, I saw the original episode with Khan, and it's a bit of a stretch to call him Kirk's arch-nemesis. He's more like "the guy who caused Kirk a minor inconvenience". The premise is that Khan is a genetically-enhanced superhuman who took over the world back in the 1990's. That was a good premise, and it set him up to be a great villain. However, the episode didn't follow through on the premise. Khan acted like a normal person, not a superhuman. He spent most of his time seducing a random woman, not plotting to take over the universe.
I preferred the 39 Clues Unstoppable series, which has a relatively similar premise. That is, the villain is a genetically-enhanced super-villain, who is plotting to take over the world. It's near-impossible to stop him, because he has super-intelligence and super-strength.
Like the episode, the Khan movie overlooks the premise that Khan is a super-genius. But unlike the episode, that's okay! Khan is still a good, interesting villain, even if he's not the genius he's supposed to be. He comes off as a legitimate threat, who has an intense rivalry with Kirk. The acting and solid script completely sold me on the big rivalry, even though, in reality, these are two guys who met each other once, fifteen years ago, not life-long rivals. Heck, Kirk even admits he hasn't thought about Khan in years, that's how little of an impact he left.
I liked almost the entire movie, except for the part where Kirk escapes from an underground trap, by using an emergency transporter which didn't get mentioned until just now. That felt like cheating, on the scriptwriter's part. But hey, one of the characters calls Kirk out on being a cheater, so at least they're aware that Kirk likes to bend the rules in order to win. Plus, the movie ends with a legitimate victory for Kirk in the battle of wits, so I'm willing to overlook the time he cheats his way out of an inescapable trap. After all, it is totally in character for Kirk to do something sneaky like that.
My wife liked almost the entire movie, except for the casting of Kirstie Alley as "Hot Mr. Spock". Or should I call her "Mrs. Spock"? Either way, it was completely unnecessary to have a sexy Vulcan in this movie, and borderline out-of-character for Vulcans to do things like try out a cute new hairdo. Spock is only half-Vulcan, and he would NEVER do that, so it's hard to picture a full-Vulcan doing it. I much preferred the random Vulcan woman in Star Trek: The Next Generation who didn't do much besides stand around, looking bored. That's way more in line with the Vulcan mythology that the original series built up.
All in all, good movie. Good writing, good acting, and the ending was extremely powerful. Apparently, the movie is really popular with non-Star Trek fans, and I'm not surprised. It's that good. Not the best movie ever, by any means--I'd rather watch Indiana Jones than this movie, any day--but it's still enjoyable and worth watching. I cannot say the same for the first movie, which deserves the condescending title of "Star Trek: The Slow-Motion Picture".