Strip all of the wires, exposing at least 1/2-inch of copper. Straighten each wire and bend it away from the others to minimize the likelihood of losing track of a wire. Loosen the screws in each terminal on the thermostat. A little bit of wire and terminal preparation now will go a long way towards making the installation easier.
Attach the red wire to the "Rc" terminal on the thermostat. This is the power terminal for cooling. Since you are only wiring up an air handler, there is no need to concern yourself with the "Rh" terminal, which is for power for heating. Wrap the red wire emerging from the wall around the "Rc" terminal and tighten down the screw on the terminal until the wire is secure.
Attach the yellow wire emerging from the wall to the relay indicated as "Y1" and if you have a blue wire, attach it to the terminal indicated "Y2." The "Y" terminals are for the compressor relay on your air conditioner and second-stage cooling. Attach the wires to the terminals and tighten down the screws until the wires are secure.
Attach the green wire emerging from the wall to the terminal indicated as "G." Not all air handlers have a green wire, so if you do not have one it is not necessarily a problem. The green wire is used for powering the indoor fan, but is not present in all thermostats.
Attach the brown wire emerging from the wall to the terminal indicated as "C." Not all air handlers have a brown wire, so if you do not have one, it is not a problem. The brown wire is used as a neutral wire for some electronic thermostats, but is not present in all systems.