Comment by salamanderman

As a start, I recommend reading about Winding Numbers. I've found that to be the most important topology concept for 2d geometry. Computational Geometry is probably deeper than you want to go, but it would answer most of your questions. For curves, I recommend a Numerical Methods reference. Honestly, wander around Wikipedia links starting from topics like winding numbers and spatial partitioning, and you'll probably get a better primer than any book.