How to start learning knot theory?

If you're willing to work on an intuitive level, you can get pretty far with very little background. Colin Adams' The Knot Book is an excellent and elementary introduction at this level.

If you want more rigor and depth, you'll want to learn some point-set topology and maybe a little group theory and algebraic topology. If you don't have a solid undergraduate background in mathematics this will take awhile and I don't recommend it if you have just a passing interest in knot theory. Knot theory gets very deep. The modern theory has unexpected connections to various fields, and to appreciate these you'll want to learn some abstract algebra leading into some representation theory, among other things... but just read The Knot Book first.

Once you have some basic exposure to point-set topology and group theory, Crowell and Fox's "Introduction to Knot Theory" is surprisingly nontechnical and intuitive. It develops most of the needed algebraic topology in it only as far as necessary to understand some basic knot invariants. It is only 146 pages and will allow you to compute basic knot polynomials.