I want to create a circuit which will wirelessly drive a relay. I only have a little experience with Arduino. I know what peripherals I need, but I don't know what micro-controller to use. I made a list and I need to know if the components will work all together with the microchip PIC18F26K80 microcontroller, or if you have any other (cheap) suggestion.
Well you can use a much cheaper PIC, like the 16F88. RF transceivers are not "that" easy to use, you could try the HC-05 bluetooth modules : they're simple "transparent" bridge, and you communicate with them through a UART (16F88 has 1 UART, and it's real easy to set up) and Arduino has one too.
You should consider buying a microchip PicKit3 instead of a cheap eBay icsp programmer. PicKit's firmware is regularly updated and programs virtually any PIC.