Where do people learn about things like caustics, evolutes, inverse curves, etc.?

Solution 1:

The National Curve Bank is a useful resource. Here are some old but still useful books on the topic:

• A Catalog of Plane Curves, J. Dennis Lawrence, Dover Publications, 1972.
• A Book of Curves, E.H. Lockwood, Cambridge University Press, 1967.
• Curves and Their Properties, Robert C. Yates, The National Council of Teachers of Mathematics, 1974.

The cover below shows a nephroid, named after its kidney shape:
     

Solution 2:

Apart from the beautiful references that Joseph has already given, I want to point out the beautiful book The Advanced Geometry of Plane Curves and their Applications by Cornelis Zwikker. In this book, the machinery of complex numbers is used to easily deduce the properties of curves, as well as derived curves like the evolute and the pedal.

Joseph has already mentioned Lockwood's fabulous book, but there is one thing I wish to emphasize about it: it is meant to be an interactive book, where you should have a number of drawing implements (or a computer if you're more inclined to draw with that tool instead of pencil and paper) on hand to fully appreciate the book.

Two more books that I have found to be nice are C.G. Gibson's Elementary Geometry of Algebraic Curves and Elementary Geometry of Differentiable Curves; the second book, for instance, has a neat discussion of the connections between envelopes, orthotomics and caustics.

Finally, if you are Francophone, you might wish to look at the wonderful Mathcurve site by Alain Esculier and others; the animated GIFs demonstrating properties of curves in that site are a sight to behold.