Remove the old thermostat. The thermostat likely has a cover that can be pried off using a screwdriver. Once the cover is off, look for screws that tie the thermostat in to the wall. Take a picture of the location of wires inside the thermostat before you remove it, and remove the screws. Remove the thermostat and disconnect the wires from the terminals.
Straighten the wires you have removed from the old thermostat. Make sure they are all accounted for and attach a clamp or vice grip to them where they meet the wall to prevent any from falling behind the drywall. Strip each wire so there is at least 1/2 inch of copper exposed on each.
Attach the red wires to the "Rh" and "Rc" terminals. The red wires supply power to the heating (Rh) and cooling (Rc) systems. Press the wires in to the terminals and tighten down the screw. Test that the wires are secure by pulling on them and making sure they do not slip out of the terminal.
Attach the yellow and blue wires to the "Y1" and "Y2" terminals, respectively. The yellow wire is tied in to the "Y1" terminal, which is for the first stage cooling system. The blue wire is tied in to the "Y2" terminal, which is for the second stage cooling system. Tighten down the screws on each terminal and ensure the wire is secure as you did with the red wires.
Attach the white and black wires to the "W1" and "W2" terminals, respectively. The white wire ties in to the "W1" terminal which is for first stage heating, and the black wire ties in to "W2" which is second stage heating.
Feed the wires behind the drywall so the thermostat can sit flush against the wall. Secure the thermostat using the screws that came with the device. Power on the heating and air conditioning system and test the thermostat by cycling on the heating and air conditioning systems.