I agree with Gowers 100%. I would only that my goal is always to find my own easy and obvious proof and avoid all trickery and sophistication as much as possible. When I run into an obstacle that I can't seem to avoid no matter how hard I try, it is often because a new idea, trick, or technique really is needed. It is only then that I peek at the reference. If I've already been beating your head against it long enough, I'm usually able to locate and understand the critical point far more easily than if I were just reading the proof from beginning to end (which usually just puts me to sleep).
What I've found is that a surprising number of theorems can be proved using an approach that is easy and obvious. Most of the others can usually be proved using a straightforward approach but using some novel idea or trick at only one or two critical steps. I find that once I see the proof in that form, I find both the theorem and proof very easy to remember.
There are, however, always some really useful theorems, where the proofs are difficult and not very enlightening. Those you end up memorizing only because you use them often enough.
I don't see the point of trying to memorize a theorem before you use it a lot.