Can I buy just any piece of electronics and be sure it will work with my Arduino?
No.
Can you give me the basic steps for generating a sound? (like a synthetizer)
Sound processing from:-
http://interface.khm.de/index.php/labor/experimente/arduino-realtime-audio-processing/
http://www.arduino.cc/playground/Main/InterfacingWithHardware
Look under Audio - although this processor is not the best one for audio due to lack of memory.