Background needed for a simple RF transmitter/receiver pair
I'd suggest you find an audio AM or FM transmitter receiver combo, and test it with audio. AM is probably easiest. Plenty of circuits available on the web.

Next the transmitting processor must supply the audio, using for instance FSK. At the receiving side you can use a PLL to decode the signal, google XR2211 or NE567.

Once you have an audio radio link, the problem is much like the good old telephone modem, so you can google for keywords like "FSK modem" or "300 baud modem". Or, google for casette tape storage - that too is almost the same problem as yours.