How to hook up the C wire from my thermostat to the air handler?
That looks fine.  

The gist is that a transformer has two wires with 24V between them.  The R wire goes to one, the C wire is the one which is not R. 

You've got a lot of exposed bare-wire 240V power in there; be careful  and shut off every furnace breaker before going in there.   There is likely more than one.