I'm going to dig into spoilers, because my only real problems with the movie (and they're pretty big problems) have to do with specific plot points and how they play out. The acting, the dialogue, the action sequences, are all top notch. It's fun, it's exciting, there's intelligent themes that are thoughtfully explored... there's a lot of stuff to like. It's abundantly clear that everyone involved with this movie was both intent and capable of making a truly great popcorn movie, one that we'd put on the same footing as The Dark Knight, Skyfall, etc.
Which begs the question of why they made some colossally stupid decisions around particular elements of the story. Spoilers abound from here on out.
I actually don't think it was a terrible idea to have Khan in there, but the writers never decide if they want to innovate like they've done with a number of other elements in the reboot or just use the character as an excuse to make a ton of callbacks to Wrath of Khan. I get why you'd want to do that - Wrath of Khan is the best storyline in all of Star Trek, but Kirk hasn't betrayed Khan yet in this timeline, so the motivation for Khan to want to destroy the Enterprise seems a bit flimsy even if it leads to a fantastic action sequence. When Khan announces his true identity he practically hisses it into the camera as if it has some special significance... but it doesn't yet, and there was derisive chortling in the theatre both times I saw it. I watched it again today to see if I had missed anything, and rewatched Wrath of Khan, and the whole sacrifice thing is just horribly played fan service. We know Kirk's not going to die so there's absolutely no tension.
The most frustrating part in all this is that Abrams can wring the emotion out of the big sacrifice scenes like nobody's business. The first 10 minutes of Star Trek do it perfectly. I cried in the theater, and it's because the stakes are life and death and while the viewer doesn't really have an attachment to any of the characters, it becomes immediately clear that they have very very strong attachments to each other that triggers that empathy response. We know George Kirk can die and that if he does, he's not coming back thanks to magic blood. His death has implications. When Kirk "dies" in Into Darkness, the only thing it triggers is another unearned callback to Wrath of Khan.
It's not unusual for summer movies to be this stupid, but it is unusual for a movie with so many other things going for it to have lapses this massive in storytelling logic and the management of audience expectations. Including Khan was fine, and changing the relationship between him and Kirk made sense within the story... it's only when they tried to shoe-horn elements from the previous movies in without the proper context that it falls apart.
And magic blood that can cure anything is just lazy screenwriting. They could have raised the stakes without making them mortal for characters we know can't die.