I just need a few book recommendations for studying on my own. I know the basics (trig, calc, etc.) and on my free time, I studied multivariable and vector calculus, in addition to differential equations. I am now trying to go into the fields of differential geometry/manifolds/topology, etc. Can you guys recommend me some books on my level? If you need me to tell you further what I already studied, please comment. Thanks in advance.
There are books dealing with the "classical" differential geometry from a non-riemannian perspective.
The keywords here would be "principal curvature" and "shape operator" - try to understand the basics of these concepts from the wikipedia and find some books (google books search) that explain them systematically and that match your own background.
Such books will likely not start with a modern approach and might even leave it out completely. Words like "elementary" in the title and keywords such as "curves" and "surfaces" often hint to classical approaches.
I have found that many lectures (notes available online) go for the modern approach straight away, most likely due to time constraints for the lecture. This would be missing a lot of the practical applications (or only getting them in a way obscured by abstraction).
Classical is fine for starters, if you have the time. Once you have understood all relevant concepts there go for the modern / riemannian approach.
Sadly I cannot recommend any specific titles, as I got my own introduction from (older) non-english books and there are likely better ones out right now.
With a good background in multivariable calculus, you might be well-poised to read Milnor’s “Topology from the Differentials Viewpoint,” or Guillemin and Pollack’s “Differential Topology,” which is more-or-less an expanded version of Milnor’s text (but also contain several additional topics). As the names suggest, these are differential topology books. I think that the only technical tool you need — but might not know well — is the implicit function theorem. I really like these books a lot. On the other hand, these books emphasize topics which seem not to be emphasized in more “modern” courses.
"An Introduction to Differentiable Manifolds and Riemannian Geometry" by Boothby seems to have a nice sampling of more “modern” topics, including Riemannian geometry, but it is a little more abstract than the above two books. I personally really like Lee’s “Introduction to Smooth Manifolds (2 ed.)”, but this might be too much for your background (or it might not be!).
For basic Riemannian geometry, do Carmo has a popular book on differential geometry of curves and surfaces (I have not read this). I like the more advanced "Riemmanian Geometry" by do Carmo, as well as the similar "Riemannian Manifolds" by Lee, but these roughly have the topics from the books in the last paragraph as prerequisites.
If you’re lacking in prerequisites for some or all of the above, it might be useful to first study a book in “advanced multivariable calculus”. One popular but terse option is “Calculus on Manifolds” by Spivak. Another popular option is “Analysis on Manifolds” by Munkres (I have not read this one). I personally think that “Advanced Calculus of Several Variables” by Edwards is quite good, and it contains more details than Spivak, although there are some errors in the text.