The fan provides most of the thrust of the engine, by far, and it does spin rather fast in flight.
Most of the air on a modern high-bypass-ratio engine goes outside the core (the bypass ratio is
the mass ratio of bypass air to core air). It is not for cooling though. It does make the engine quieter, but the reason for bypassing the core is that it brings the engine efficiency up tremendously to accelerate a lot of air a little, as compared to accelerating a little air a lot.
However, a lot of the air which goes through the core (gas generator) does not take part in the combustion process but is used to cool burner liners, turbine etc. Getting this amount of air down is one of the challenges of making jet engines today.
contains more information, dunno if it is good but it should point you further if nothing else.
I thought I was doing good trying to avoid those airport hotels... and look at me now.