I wouldn't agree with your interpretation of free groups, I think that your understanding comes from the free group on 2 letters with a particular action on a fractal. That's not to say that there isn't much of interest to say about free products and actions on particular spaces.
But if you're just interested in free groups and how they are related to free products then I'd recommend reading up on Bass-Serre theory: http://en.wikipedia.org/wiki/Bass%E2%80%93Serre_theory
This studies how groups act on simplicial trees.
Current research is focused on the automorphisms of free groups, if you want to learn about this you should search for articles on "Outer space", possibly by Karen Vogtmann.