Once I get over all the things left out between the book and the movie (always a wrench), I can enjoy this movie a great deal. It's darker, although perhaps not darker than the Order of the Phoenix. It's also more involved with the personal lives of our main characters. Oddly, though, the kids have grown up so much that I can't be sure which ones are or are not being played by different actors. Not the main characters, certainly: but is Lavender Brown the same? Neville and Luna may or may not be: they are seen so briefly.
Anyway, the basic story is that the Death Eaters are causing a lot of trouble, even in the muggle world, and Dumbledore seems to have a plan to fight back, although the nature of the fight takes some time to develop. Even for him.
He has asked Horace Slughorn to come out of retirement and be the Potions master, while Severus Snape finally gets the position he has wanted all along: Defense Against the Dark Arts. Why? It turns out Tom Riddle was a favorite of Slughorn's when he was at Hogwarts, and Dumbledore is convinced he has vital information about Riddle. You see, Slughorn is a sort of name-dropping groupie of famous witches and wizards and their children. As Dumbledore puts it, he likes to "collect" names for his Slug Club. Of course he wants to collect Harry, and Dumbledore asks Harry to get close to him in the hopes of getting the necessary information. Dumbledore has a bottled memory of Slughorn talking to Tom Riddle, but it has been altered. (You know about the bottled memories; they are put into the pensieve in order to be experienced.) It is Harry's mission to get the truth from Slughorn.
Harry succeeds. It turns out Slughorn told Riddle about horcruxes, very dark magic that allows a magician to divide part of his soul and hide it in an object. So if he is killed, it's just that body that dies; he can still recover the hidden part of his soul from the horcrux. Making just one is bad enough, but apparently Voldemort made seven of them. Potter destroyed one, Riddle's diary, in the Chamber of Secrets. Dumbledore found and destroyed a ring which had been in the family of Riddle's mother for some time, and it blackened his hand in the process.
Now he believes he knows where another is hidden: in a cave by the ocean where he takes Harry with him, first extracting a promise from Harry that he will do as Dumbledore asks, no matter what. Harry promises, but it turns out to be pretty horrible to live up to. And even more horrible is what they confront when they return to Hogwarts.
I've told you too much already, especially if you haven't read the books. But believe me, there's a great deal more. For example, just what is Draco Malfoy up to?