work (one's) way through (something)

1. To remain continually engaged in some task. Often used when the task is long-term, tedious, or plodding. We're working our way through the set of problems the math teachers assigned. Even if I'm not really enjoying a book, I feel like I have to work my way through it once I've started it.

2. To work in order to pay for some long-term educational program. Kate is working her way through college, but it's taken a while, as she's only been able to attend classes part-time. You don't even realize how well off you are. When I was your age, I had to work my way through school!

work, etc. your way through something

read or do something from the beginning to the end of something: He worked his way through the dictionary learning ten new words every day. ♢ He’s eating his way through all the restaurants that are recommended in the Good Food Guide.