denno:
Where could I find exactly which crystal and capacitors I would need? Being able to program the chip whilst it's sitting in the Uno board, and then just taking it out after that would be the most simple way I would think, at least until I get a better understanding of how the circuitry works.
Unless you need good frequency accuracy (which you don't, because you are using a RTC) then the simplest solution is to use a 8MHz or 16MHz 3-terminal ceramic resonator, which replaces the crystal and the caps.
denno:
What would I need a reset button for? Would that be as simple as adding a push button to pin 1 of the chip?
You don't need a reset button, you can reset the chip if you need to by disconnecting and reconnecting the power.
I recommend including a 6-pin ICSP header in your design, so that you can program the chip in-situ. See Prototyping small embedded projects with Arduino | David Crocker's Solutions blog for more.