To learn the basics you need to understand in a reasonable level classical mechanics, quantum mechanics and special relativity. The conformal field theory you need will be presented on the string theory textbooks. Actually even the basics of special relativity and quantum mechanics is presented in Zweibach's "A first course in string theory". Once you have understood the quantized string you will want to understand the spectrum. For this you will need a lot, but a lot, of group theory (including how to decompose spinor representations for different dimensions). Then, you will want to understand current algebras and thus a lot of CFT. Later you will need to understand BRST quantization (thus some elements of cohomology), D-branes etc. To understand string amplitudes you might as well need some alg. geometry and the story gets more complicated...

The best thing to do is to follow Susskind's "Theoretical Minimum" which should prepare you well to understand some basics of string theory, but it is definitely hard to understand well string theory without **solid **fundations in classical and quantum mechanics, special and general relativity and of course quantum field theory. If you want to professionally work on the area maybe it would be a good idea to do a Master's in theoretical or mathematical physics first. It is possible. When I was doing a masters one of my classmates was a chemical engineer and he excelled in the master (after many many hours of work though). As for the Hatcher and algebraic topology it wont really help you for a while. I cannot remember from the top of my head now if homotopy is really needed in any of the classic textbooks, and the only thing usually needed is some elements of cohomology (de Rham, BRST etc).

Good luck.