I'm no longer in academia and while my job has some mathematical challenges, they aren't as interesting as the challenges I find in other branches of mathematics. That means I do mathematics on my own as a hobby.
As I see it, the main challenge is this: how do you know you understand? You can try doing the exercises in textbooks. But how do you know you have the correct solution? And if the book has solutions, how can you trust the similarity metric you use to decide whether your solution is the same one? It seems to me that from time to time you need external calibration to make sure you're on the right track. In a sense you have to always test yourself, in the sense of falsifiability. If you work in academia, your colleagues will keep doing this for you. I know two ways to achieve this on your own:
(1) Blog about what you have learnt. Because of the "someone is wrong on the Internet" syndrome, you're likely to get a response if you say something that is incorrect. You can write page after page of insightful material that will (apparently) be completely ignored, but if you make a mistake you'll get corrected (at least if you can get a following of some sort). As a side effect, communicating stuff to other people, even "rubber ducks", can really deepen your own understanding.
(2) Try to turn what you have learnt into computer programs that solve a problem. Computer programs don't provide proofs (well, that's not always true), but they will give you confidence and sanity checks. This doesn't just limit you to numerical analysis. Whether you're doing group theory, or algebraic geometry, or logic, or algebraic topology, or combinatorics, or even analysis, there are often subdomains of those disciplines for which you can write computer programs that will likely fail if you don't understand the theory.
