The Doctor intends to travel to some holiday planet, but naturally ends up somewhere he didn't want to be. So far so good. Then Amy goes back into the TARDIS to get her camera even though they're clearly in the wrong place, and the Doctor and Rory wander off instead of waiting five seconds for her to get back.
Obviously this results in them being split up, but they find a way to talk to each other, so they
It turns out that Amy has been quarantined because they think she has a plague that only infects species with two hearts. How the fuck does that work? Is every two-hearted species related? And not only did this medical facility not check to see if she actually was infected or just pressed the wrong button, but it hasn't even bothered to check her species. It does scan her for bacteria though and decides that she's infected with a whole bunch of other foreign bacteria, and tries to cure her, which would kill her. Worst medical facility ever.
So the Doctor and Rory go back to the TARDIS and try to get to where Amy is, but end up being 36 years later than they'd planned. Fortunately Amy is still alive, but despite not needing to eat or sleep or anything, she has aged. But judging from appearances, not 36 years. This is never addressed.
As well as aging, Amy has apparently gained a whole lot of new skills as she's managed to build her own sonic screwdriver and reprogram the facility's computer. And she has a sword. How the fuck did Amy do any of this? Amy doesn't know how to do any of those things! And where did the sword come from? Also, somehow Amy knows that if the robots touch you, you fall unconscious. But if that had ever happened to her, she'd be dead, and there's no one else around, so how does she know?
And while I'm on the topic of other people, why are there no other people around? Every single patient seems to get their own personal timestream, so none of them ever meet or interact with each other. What the fuck? They've all got the same disease, and you've somehow made it so they can live out their normal lifespan despite having it (presumably the same magic that made them not need to eat), so they could all freely live together, no problem, but instead you've got them in solitary confinement? What the fuck? Worst medical facility ever.
Fortunately you can alter your own past this week, so with Amy's help they can go back and save the past her, but she won't help, because she's fucking insane. Not insane like you'd expect someone to be after 36 years of solitary confinement, mind you, just regular "I'm in a shitty TV show" crazy.
The issue is basically that she thinks that changing her own past is somehow equivalent to dying, because past Amy will do different things so she won't exist as she now does. This makes a sort of sense if you don't actually think about it. If you go back in time and warn yourself about something bad that happened to you so you can avoid it, is that anything like dying? No, it's not. You still exist, you just didn't have that bad thing happen to you.
So the Doctor has to trick her into thinking he can save "both" Amys, and they do the big escape. Then the Doctor locks her out of the TARDIS and Rory's all like "You're just letting her die!" because he can't seem to understand the simple concept that there is only one Amy. You can't save both Amys because there aren't two of them. There's only one, and you can either rescue her a few days after you lost her or 36 years after. If you travel to yesterday, there seem to be two of you for a day, but it's actually only one person, the earlier you just becomes the later you.
Oh, and the other annoying bit was all the talking Rory up that Amy did. God, he is a pathetic doormat with no personality. There is nothing interesting or likable about him. He barely even exists as a character, and what personality does exist is terrible. And as with the flashback in Let's Kill Hitler, Amy talking about her past just makes it clear that she is a terrible, obnoxious person.