Unfortunately, I picked electrical engineering instead of pure maths courses in college and now I can't switch back, but there's still a chance for me to become a pure mathematicians by attempting the GREs. Here's my reading list uptil then:
1)multivariable calc(already done)
2)linear algebra (already done)
3)Ordinary and partial differential equations (studied insofar as it benefits an engineer)
4)abstract algebra
5) Analysis (Rudin, this is where I am right now)
6) I guess more analysis, with going into details about measure theory and functional analysis
7) complex analysis
8) topology
9) more topology
10) even more topology(algebraic topology)
11) differential geometry, but not the elementary course
12) representation theory, with picrel as introduction. Will continue until I have enough knowledge to decipher advanced QFT stuff

Please consider the fact that there is no dearth of time, because I'm basically devoting my entire life to maths and I wanna be able to read PhD level papers on very niche fields with relative ease. This list will serve as just my introduction to maths.