How should I go about reading mathematics papers and textbooks as a PhD student
I'm a second year student too.

I think that one have to study what really needs to know, for instance, if I need the Feit-Thompson Theorem (for some reason) I cite and use, if you need to use the tools and ideas that were used in that long proof, then you read the paper. Now, how to read it, in first instance, keep the central ideas and go deeper if you really need it for your work. There is a lot of mathematics, and nobody can learn everything, even if you only study algebraic geometry (it is a broad field). My tutor says that one will have a lot of years to study and go deep in topics that are of our interest, and in the moment, work hard for getting the PhD (it is not easy, we know it), and publish