So, I'm currently at the point of learning about derivatives and integrals in Elementary Real Analysis, I'm also learning Point set topology and I've already gone over the main definition of topology, basis, subspace topology, quotient topology, product topology up to continuous functions thus far. I also understand elementary linear algebra and know enough Abstract Algebra.
My question is whether or not I can start reading Differential Topology by Guillemin and Pollack. It states a full year of analysis and semester of linear algebra but I'm not exactly sure if my current level understanding is sufficient to start studying differential topology.
If not should I start going over Hatcher's book on Algebraic Topology? I've read that knowing algebraic topology before differential topology is a good idea.
I would say no. In my experience, in order to really study differential topology you need to have a firm ground in multivariable calculus. In particular, things like understanding the derivative is a linear map (best linear approximation), implicit function theorem, inverse function theorem, etc.
Ted Shifrin has a really awesome book that will give you the relevant linear algebra and multi- calculus to begin studying differential topology here. I would also recommend Tu's introduction to manifolds, or Lee who develop a bit more geometric understanding of tangent space (than G & P) which is the first topic to digest. A lot of users may not like this but I think the order of things follows best if you go,
$$\textbf{linear algebra + multivariable calculus} \to \textbf{differential geometry} \to \textbf{differential topology}$$