If I were to develop my own radio device, what frequency(s) would I need to use? I can't just use any old one because they are allocated by the FCC.
This depends on which country you are in and what kind of application the radio is for.
Since you mention FCC, I assume USA. FCC part 15 lists all bands that can be used without a license, given that you follow the restrictions for the given band stated by that document. Which band to pick depends on the nature of the application.
For generic short-range devices, you could look at the 900MHz band. It is not an easy band to use though, as you need to utilize frequency hopping, or alternatively send with very low power. Also, this band can only be used in the US, Canada and a few other countries that accept the FCC regulations.
There exists no band which is completely free of regulations. Generally, no matter band, you will need FCC approval for the finished product, which is obtained from tests at an FCC-approved test house. Additional tests/paperwork is needed for use in Canada.