If the switch is off, do I need to also turn off the circuit breaker? Is there any power to the light fixture when the switch is off?

Could be, depending on how it was wired. If power is through the switch, shutting off the switch will stop the power. If powered through a switch loop, there will always be power at the fixture box, even if the switch is off. Look at the switch. If it has 2 black wires it is most likely hot through the switch. If you have a black and a white or a black and a white with black tape or marked on the white, you most likely have a switch loop.