What size breaker do you need for two 60W bulbes 110 volts?

It is not the number of bulbs that you worry about. It is the wire size that is your concern. If the circuit is wired with AWG 12/2 wire then use a 20 amp breaker. If it is wired with a AWG 14/2 wire then use a 15 amp breaker. You are protecting the wiring with the correct size breaker.