I asked the ##math chat room on Freenode about cohomology recently, more precisely I expressed that it was for me the scariest math-word-that-I-don't-know-yet, and I received the explanation "it isn't that scary once you get used to it; it's just a way to repair exactness of sequences." Maybe that helps?

For topology, I feel like there's a sort of meeting-of-two-different-things; one starts with being very frustrated with the delta-epsilon-definition of "limit" and its one-dimensional nature, and the limit-definition of "continuous" and its clumsiness. The other starts from wanting to play with spheres and Möbius strips and knots and the like. When you're playing with these shapes a bijective mapping between two surfaces is not a fine-grained-enough idea because it is not continuous; adding continuity gives "homeomorphisms" which also aren't a fine-grained-enough idea because they do not make reference to the space an object is embedded in; wrap a torus about itself in a pretzel knot and you have something which is homeomorphic to a torus but in 3D you can't get there without tearing part of the surface through the other one, but in 4D you can. So finally we come to the idea of an isotopy, which bumps "continuous" to the next level by saying "Just like you can have a continuous path of points in space, you can have a continuous path of homeomorphisms from one to another," and that's where the pretzel knot becomes finally distinct from the torus in 3D, there is no continuous path from the homeomorphism of the pretzel knot to the torus, to the identity homeomorphism of the torus to itself. Or something like that. So this path is then an "isotopy" and then certain things are nicely isotopy-invariant and so forth.