If you connect an electric generator to an engine via shaft and
assuming the engine runs at 2500 rpm than the electric generator will
produce electricity at the rate of 2500 rpm, if you increase the speed
of the rotation by using gears than I assume the generator will make
more electricity. What would be the factor limiting at how fast we can
turn the generator? i understand that some of the steam and gas
turbine has a gear to reduce the speed of rotations of the shaft
leading to the generator, why can't we use the full speed of the
turbine to spin the generator at the same rate, and by not using the
full potential of the turbine is it less efficiant that way?

Your assumption is wrong. You would actually generate less power due
to friction losses in the gears. When you increase the speed, you
decrease the torque. The power remains the same, minus the gear
losses.