What is the best way for me to get my cool project off of the Uno and onto my circuit board?
You could use the Uno as a programmer:http://arduino.cc/en/Tutorial/ArduinoISP
(ignore the part that says the Uno doesn't work for this - it's outdated)
Can you point me to some directions which list the components necessary to run a ATmega328 based circuit?http://arduino.cc/en/Main/Standalone
Note that you may not need all of the components; e.g. If you're using a specific, regulated power supply you may not need the 7805.