The levels progress in accordance with how the horse should be trained. So at intro level, we are really just getting the horse accustomed to the arena, and want them to go forward in an unrestricted manner, with a soft acceptance of the bridle. We then gradually move up the levels, and are expected to have the horse more engaged, more responsive, and begin to work in collection. The lateral movements are gradually introduced, beginning with leg yield, moving the shoulder in, travers, half pass and so on. Counter canter, simple changes etc.
Finally to Grand Prix, where we have piaffe, passage, one tempi changes, canter zig zags and so on.
At each level that you go up, the complexity of the test increases slightly. Each one rides differently, and it also depends on the individual horse's strengths and weaknesses as to which tests they find difficult or easy at each level.
As we move up a level, the movements are performed closer together, with less time to prepare before the next movement.
As for what you learn in lessons... well. You learn to ride basic flat work and dressage. You will learn the absolute basics, don't expect to go into a 'dressage' lesson and come out riding piaffe. You will start with getting your own position effective and balanced, moving to learning to 'feel' the movement of the horse and how and when to apply aids for maximum effectiveness. You will learn to develop a horse's training to have it swinging and through its back, into the bridle and begin to learn some lateral work, collection etc.
It is a LONG process, you never stop learning. Many people go into dressage lessons, with the impression that it's going to turn them and their horse into instant superstars. This is not the case, it's a LOT of hard work and takes years upon years. You have to have drive and dedication to stick with it. But my god, is it worth it.