I'm fairly new to arduino, but I want to program a nano over SPI to skip the bootloader. I know I can program the nano with an uno as a programmer to do this, but I don't have an uno, but I do have a couple of nanos. Can I set up one as a programmer, connect it to another over SPI, and program the 2nd?
Yes, you can. They use ATmega328 / Atmega328P microcontrollers, which are basically the same.
(The differencies of those microcontrollers are about Brown Out Detection - how they could be set to react on too low voltage. No problem in nearly any Arduino operation.)
The boards differ mainly in the board size and places for pins.