This may not make a lot of sense but:
I'm interested in studying more about topological manifolds, smooth manifolds, and maybe some algebraic topology later etc. but I'm not sure if I need to have a good prior understanding of classical geometry. My geometry sucks, period, and I know that modern geometry and topology provides us a modern way to approach problems in geometry; however, if I don't have a very good understanding of classical geometries (classification 2-d surfaces etc. and what not), I'm not sure if I'd be able to either appreciate or dig deeper into the subjects. For example, in analysis, we don't study functional analysis before linear algebra, and we don't study linear algebra before Euclidean geometry because each subject builds on the previous one.
Am I over complicating this issue? I need some suggestions on how to effectively dig deeper into these subjects.
You're overthinking it a little, IMO. You really should just pick up a good book like Lee's Intro to Smooth Manifolds and see if you can follow along. Oftentimes the classical theories are presenting specific examples and cases of the more general theory. In my experience, studying a general theory can give me a foothold for understanding something "more elementary" precisely because it provides a broader context in which special cases can be understood.
Now, all of this goes with the assumption that you've studied basic courses like analysis, linear algebra, and some group theory. If you haven't studied those subjects, then you really should gain familiarity with them before studying differential geometry. However if your concern is more regarding, as you said, "not knowing the classification of 2-surfaces," then you should really just jump in and get your feet wet.