You *CAN* throw clays yourself with a hand-thrower. I wouldn't recommend it until after you have shot clays a good few times. That is a lot to think about, and too much to think about at once can become dangerous. After you get more practice you are not quite as concentrated about the swing, mounting, leading, etc and you can concentrate on safely holding the shotgun by the foregrip in your non-dominant hand, throwing with your dominant hand, dropping the thrower and mounting the rifle.

I have seen spring throwers at walmart for reasonable prices... I can't remember how much but I think they were $50 or less, but don't quote me on that.

If you use #7.5 or #8s you don't really need a backstop. In fact, unless you are shooting into a mountain the clays will be above most back-stops. But you should make sure that no people will be within several hundred yards (the more the better) of your shooting in the down-range direction. The bigger the shot (smaller number) the further it will carry with deadly force, so stick with the small shot and get the lower velocity varieties to help minimize travel.

Get the biodegradable clays. You have to make sure they stay dry/indoors unlike the traditional clay clays, but they will degrade naturally and quickly.