I'm interested in learning a bit of geometry. To start I'm (slowly) working my way towards differential geometry via Lee's Introduction to Smooth Manifolds. But, later on, I'd also like to study some (real) algebraic geometry (I'm interest in how it interacts with optimisation).
I have a fair bit of exposure to analysis (real/complex/functional, topology, measure theory, probability theory, optimisation, PDEs, stochastic processes, etc.) but virtually none to algebra (only linear algebra) or category theory.
Last trimester I attended part of an introductory course on smooth manifolds (couldn't finish it because too many things were going on at the same time). In it the lecturer occasional discussed concepts from algebra (for example, groups) and category theory (for example, universal properties) and I felt that I was missing out.
To rectify this I've been reading Conceptual Mathematics: A first introduction to categories by Lawrence and Schanuel. However, I found this mo post which got me worried I might be going at this the wrong way round.
So my questions are:
With the end goal of acquiring a working knowledge of differential and algebraic geometry, how much algebra and/or category theory should one know?
What references would you recommend to achieve the above, is Lawrence and Schanuel's book a good start? Even if so, what else would you recommend?