I do mathematics as a hobby, and have found various blogs and MathOverflow and Math StackExchange very helpful in continuing my education and revealing my limitations.
I have always enjoyed maths, and done well, but in the end that wasn't a good enough reason for me to make a career. I became interested in too many other things as well.
So what do I do?
First, I do puzzles and problems - I enjoy solving things, and always have done.
Second, I read textbooks - the demise of bookshops in favour of online resources is a bit of a menace here, because in a bookshop I could browse more easily for something interesting which appeared to be within my range and opened up an area I might be a little unfamiliar with.
Third, I do intentional study - for example to understand the classification of finite simple groups, or (as far as possible) the proof of Fermat's last theorem, or the Riemann Hypothesis, or PvNP. But (to give a benefit of online resources) I do download a number of the papers linked in posts on this site. I did some Algebraic Geometry when I was younger and am now trying out Ravi Vakil's notes to get myself up to speed - but as a hobbyist I don't always have the time to consolidate what I've read.
What I do find is that I miss some of the informal knowledge (the stuff people talk about, but don't write down) so I don't always link things together as quickly as I might. And also I find that my intuition is not what it was - I think the exercises in good textbooks feed intuition by giving a sense of what is possible and what is not, and help to direct imagination in fruitful ways. I am not doing enough work to sustain my intuition at a high level.
However posting answers on sites like this does force me to commit myself in public, and I find that a learning experience, with lots of helpful (if occasionally sharp) comments and feedback. Since such self-learning is not in line with the research goal on MathOverflow, I indulge myself rather more on StackExchange, and tend to scan here for insights on things I might be reading at the moment.
