Also, use computers! I use SAGE extensively to get a feeling for lots of mathematical objects!

1. Grobner bases: http://bollu.github.io/computing-equivalent-gate-sets-using-...

2. Localization: https://github.com/bollu/bollu.github.io/blob/8cd335687ff3ef...

3. More broadly, an answer on math.stackexchange on how to debug math: https://math.stackexchange.com/questions/1769475/how-to-debu...

4. (WIP) continued fractions to compute pi: https://bollu.github.io/fractions/index.html

And so forth. I find the computational aspects of most theories to be very rich, and it's really gratifying to code something up and "read off" the results.