My own personal routine:
I start reading, whenever I feel there is a problem presented in the text (mostly theorems and examples but sometimes questions arise in the text itself), I’ll try solving it. I invest time in this endeavor proportionally to my belief that it can be solved in a reasonable amount of time. If this time runs out, I’ll start reading a little of the solution and trying again. I usually postpone the end-of-chapter exercises to “later” because of spaced repetition and lack of time (I have to have studied a certain amount of chapters for exams so I have to prioritize the quantity.). Sometimes interesting ideas/quasi-problems/ways of looking at things come to mind during the study, and I think on them as well.
When I’m solving a problem, I’ll usually walk around and try to solve it mentally, because I have this uncertain belief that if my math needs paper and pencil, then I can hardly use it “at will.” I’m the kind of person who forgets stuff and needs to repeatedly derive them from first principles when needed, and I can’t quite do that if those derivations rely on writing stuff out. (I also hate memorizing theorems. It feels stupid. :D )
By the way, if the solution is not in the text and I’m stumped, I websearch around. I hate the professors who give grades for the text’s problems and thus disincentiviz solution manuals economically and socially. These take-home exercises correlate more with people’s social skills and network than their effort or knowledge. Quizzing people instead is a much better choice.