For non-development topics (say, math) it's again a mixture of reading books, watching videos, working exercises, etc. And I try to attack problems from multiple angles, by using different books and resources. Just to illustrate with an example:
Last night I started working through How To Prove It by Daniel J. Velleman. I got to the Chapter 1 exercises, and hit a point where I needed to know if a given number was prime or not in order to verify my answer. It was a number larger than the list of primes I have memorized, and I didn't feel like testing all the possible factors by hand, so I just fired up R on my laptop, loaded the matlab library and used the isprime() function.
Later in those same exercises I got to a point where I had an answer, but wasn't sure if it was correct or not, so I just googled until I found a guy's blog where he posted his answers to the same problems. My answer matched his, so I felt fairly confident that it was right (of course, we might have both made the same mistake).
Had I not found that guy's blog, my plan was to post a question to math.stackexchange.com or one of the reddit.com math forums, asking if somebody could verify my answer.
So I started with a book, and eventually used a number of other resources as part of my overall learning experience (a scientific calculator was mixed in there somewhere as well, before I pulled out R).
This might lead one to ask "well, how did you learn R?" First of all, I wouldn't yet rank my R skills very high, but to the extent that I know some R, it was from (again) a mix of sources:
1. A bunch of Coursera classes, including several of the ones in the Johns Hopkins Data Science track, and the Duke "Probability and Statistics with R" track.
2. Working through parts of several R books, including R In Action, Learning R, Using R For Statistics, etc.
3. Trial and error, playing around, using the built-in docs, etc.
That's kinda it. I take notes using pencil and paper, sometimes a plain text file, occasional an OpenOffice Writer document, and sometimes a wiki (I keep a MediaWiki instance running locally).