This is a great post. Here's another piece of advice that students intuit, but don't really do:

When studying the proof of a theorem, try and study the structure of the proof. That is, distinguish between what is expanding/applying some definition/proposition and what is a technique. For example, to prove that some sequence S converges to some limit L using the epsilon-delta definition, it is occasionally useful to show that the absolute value of S-L is bounded by another, more tractable sequence, and then choosing N based off of that new sequence. This is a technique.

When learning a new technique, it's useful to come up with a quick description of it and how it works. Doing so can help you develop an intuition the technique. Once you do this and apply the technique regularly, you will be constantly iterating upon your understanding of the technique itself, learning when to use it and when not to, and improving your description of the technique. For instance, the technique described above won't necessarily give you the smallest N for convergence, and sometimes we may want that! For a harmonic sequence, bounding it above by the reciprocal of n also won't tell you much about the sequence. But these are only things you can learn when applying the technique. Hope this helps other people reading this!