For sturdier sides, there is a product out there that is like the headliner stuff in the interior of your car. I am not at home right now to look at the name on the pkg but will check later for you. It is very nice...your bag will have a nice 'feel' to it...kind of similar to what is used in Vera Bradley handbags. You can get it at your LQS.
For sturdy bottoms, I have used foam presentation board or even cardboard from boxes of athletic shoes. Make a little fabric sleeve to slip it in and nobody knows. Easily replaced, too.