Suggestions for making the most out of a set graduate school program? I was recently admitted to a US graduate program in Pure Mathematics with an assistantship that requires teaching. I begin in the Fall.

The program is designed in such a way that Fall and Spring have Analysis/Algebra and their continuations, respectively, give or take optional seminars and a teaching supplement course. In the summer following the first academic year, Qualifying Exams in these topics are required that both allow one to continue into the second year of the MS program and/or pass on directly to PhD program. Both MS and PhD have dissertations at the end of the program.

My concern is the first academic year with the courses restricted to the two A's. While definitely necessary to prepare for the QE's, I wonder how much time that leaves me to study other topics, perhaps topics that would be part of an eventual dissertation. With time split between these intro courses and teaching, which will no doubt be time consuming, how much time does that leave for personal exploration? For branching out into other areas of mathematics​? 

The immediate conclusion I drew was to read other topics. But self-directed learning moves slow for me, I'm easily distracted and life gets in the way. The classroom environment is where I learn best, to be honest.

I guess I am looking for recommendations to make the most out of the graduate program's pace - how can I build the foundation required for my program while also exploring topics that will not have explicit instruction? To explore other areas is my ultimate goal for returning to school. Suggestions?

Answers both relevant and irrelevant to math are appreciated.
For graduate programs with this kind of structure, there is usually an expectation that your core courses (algebra and analysis) will be your main focus for the first year.  So you should not necessarily expect to have much extra time to spend on other topics during that year, nor will the program be expecting you to do so.

If you find that you are able to handle those courses pretty easily and have a significant amount of extra time and energy, then you could consider:


studying additional topics on your own
enrolling in other courses, if there are some that don't have the core courses as prerequisite
approaching an individual professor in an area of your interest about a reading course or independent study


But if not then don't worry about it.  There will be plenty of time after your first year to explore other areas of math, and having a thorough grounding in core areas will make that exploration much more productive.