How to learn linear algebra
It made me
moderately competent 
"Each time I came back to it, it made a little more sense."
How I did it: My first linear algebra course was as a freshman, and although I muddled through the problems (many proof-based), it still didn't really click. Or, what clicking moments there were flitted away in the following weeks and months.
I came back later when I realized I need linear algebra for quantum mechanics. When I studied it again with a little more mathematical maturity and a little more patience, I started to see that matrices are arbitrary collections of numbers with strange rules, and that linear algebra describes real physical things.
Lessons & tips: Learn both theory and applications. Go slowly. Do problems. Think up your own problems. Explain things to people.
Resources: Gilbert Strang's lectures on OpenCourseWare were invaluable (http://ocw.mit.edu/OcwWeb/Mathematics/18-06Spring-2005/CourseHome/index.htm). Sheldon Axler's "Linear Algebra Done Right" helped me a lot from the pure math side.
For the very basics, here's my attempt at a humorous lesson about what a vector space is:
