niq_ro have shared his Arduino based FM radio design on insrtuctable. TO catch FM waves he used TEA5767 FM chip which can be interfaced to Arduino via I2C interface. To adjust volume he used MCP4261-502E/P – dual digital potentiomer controlled via SPI on 8 bit steps.
He made some effort to implement pretty nice interface on Nokia 5110 display. Here you can see the frequency of band, volume level, signal strength gauge. The only thing is left to do is to put everything in to enclosure.