I prefer to be told "what's really going on" before practicing computations; this is both more intrinsically pleasant (I find) and aids in memory. See my post on Bayes' theorem, where I contrast my abstract approach with the usual one, which starts with concrete examples.
Perhaps an even more extreme example would be multivariable calculus: I was never able to properly remember, let alone understand or apply, the theorems of Gauss, Green, and Stokes until I had learned the formalism of differential forms and the generalized theorem (itself arbitrarily called Stokes' theorem, although either of the other names could also have been used). Presumably this is avoided in a first course because the idea of a multilinear map varying from point to point is considered too abstract -- a completely spurious reason in my case.
There is a widespread false assumption that computational facility is a prerequisite for theoretical understanding, when in reality they are independent skills. An unfortunate consequence of believing this falsehood in the case of someone to whom theoretical understanding comes easily (e.g. me) is that you can get the impression that you don't need to bother training computational facility (after all, you're "already" doing the thing for which it is supposedly a prerequisite), and even eventually end up thinking that doing so would be beneath your status.
What would be ideal would be a larger version of Axler that supplemented the theoretical exercises with large numbers of computational ones (Axler itself contains a few, but not enough).
Hmm.
I think learning styles differ here. The thing that helped me understand multivariable calculus was taking it concurrently with electromagnetism. I really liked having a concrete example to have in mind whenever we talked about vector fields -- most of the theorems about general vector fields have specific physical consequences, and I find remembering the physics helps me remember the theorem.
This will not be a long post; I have a simple question to ask: if you wanted to educate yourself to graduate level in mathematics, but didn't actually want to go to university, what would you do? I would ask for text-book recommendations, but I don't want to limit your responses (however, bear in mind that the wikipedia articles on, say, cardinality or well-ordering go over my head – they may skim my hairline, but over they go). Also bear in mind that while I personally have A-levels (British qualifications) in both Maths and Further Maths (which is to say, I know some calculus at least), there are probably plenty of people on lesswrong who don't and who desire the same information – so assume as much ignorance as you feel necessary (it's a shame, actually, that there isn't a sequence here on lesswrong for maths). What do you advise (if you think the query ill-defined, I would like to know that as well)?