What drives the compressor before the turbine provides power? I understand that the power extracted from the turbine is used to drive the compressor which sucks and squeezes air sent to the combustion chamber, and the cycle continues.

But initially the compressor has to suck in air! What drives it at this stage before the turbine takes over?
Turbine engines are spun up to idle speed using a starter.  This is typically in the form of an electric motor for smaller turbine engines and APUs and pneumatic powered starters for large turbine engines.  Once the gas core is spun up to a minimum speed, typically around 12-20% maximum core speed, fuel is introduced and ignited, causing a self sustaining combustion and the engine core spins up to its idle speed, typically about 50% - 60% maximum speed, before the starter cuts out.  Care has to be taken not to introduce fuel to the engine prior to reaching these speeds as the engine can overheat during startup and be seriously damaged (this is about the only disadvantage of aviation gas turbines).