Dan Murfet has some notes on foundations for category theory which can be found here. They contain an introduction to Grothendieck universes as well as some references for learning about NBG class theory.

If you are particularly interested in some more possible foundations and their pros and cons you might want to have a look at this blog post by Kenny Easwaran.

I think that if you only want to learn introductory category theory, especially for the purpose of doing homological algebra, one can often safely just pretend that size issues are not really issues. It is true that there are many technicalities involving size that can trip one up. However, I tend to see these as "just" technicalities especially from the point of view of how I think about category theory. Maybe this is a controversial point of view?