Yes, there is a rule. The verbs conjugate depending on who is doing it:
GIOCARE = TO PLAY
io gioco = I play
tu giochi = you play (s)
lui/lei gioca = he/she plays
noi giochiamo = we play
voi giocate = you play (pl)
loro giocano = they play
The c becomes ch before i so the pronunciation remains /k/ and does not become /ch/. This is a universal Italian spelling rule and is not specific to verbs.
Regular Italian verbs follow a conjugation pattern depending on how the infinitive ends.
I do not know when to use the first -ire conjugation and when to use the second -ire conjugation, though.