When the chain is worn the cassette and chainrings are also likely to be worn, especially if you drove around that way for a while. Parktool has an illustration for how to tell when the teeth are worn.
I don't know if it's worth it to seek out a bike co-op for a cassette or freewheel removal tool. It's not much more complex than a socket.
Shifters can easily be replaced.
I have used nashbar branded cassettes that held up pretty well. You don't need to spend a fortune on a cassette.