I want to emphasise something that was briefly touched in other answers, but which I think is the core of active learning.
Taking notes and reading on your own is, in my opinion, still quite passive. In fact, mindlessly taking notes during lectures can actually be a disengaging process. The only way to really learn actively is by actually struggling with hard problems.
Now, this is something many people end up passing on. I have many colleagues who just go to the lectures, do a couple of mandatory assignments, nod at the teacher, but when it gets hard they just give up (and often come ask questions here...).
Now, learning by struggling with hard exercises is awfully time consuming. You can easily spend 10+ hours more per chapter in a book like Munkres or Rudin, but at least you'll be able to actually use what you've learned actively in the future.
For me the best reference for training this kind of learning are books like Putnam and Beyond, which pose very challenging questions related to most major undergraduate topics. Unless you're extremely gifted, working through the first chapter in 'basic techniques' like induction and proof by contradiction will leave you feeling sorry for your own stupidity. But when the solution does come it is usually very rewarding.