You never stated what furnace model you have.
You should have marked the low voltage wires or noted their location on the existing thermostat before removing the wires. Look at the wiring diagram(s) of the new thermostat before starting (making sure it is suitable for your application. Turn off the power to the furnace & AC unit. Note low voltage wiring diagram of the furnace.
Is there a small 1 or 2 amp fuse or a resetable circuit breaker on the 24 volt transformer, 0r in low voltage circuit. Note low volatge wiring diagram (this would be shown in it)
Does the furnace have R,W,Y,& G terminials?
Remove front panel of furnace and trace exsiting low voltage wires from thermostat to the circuit board terminials. Find the two low voltage wires going from the furnace to outside compressor unit (AC) One wire will be on the C terminial to other wires will be on the Y terminial. There is a second wire on the Y terminial that will go the Y terminial on the thermostat Note the color of this wire. Now connect this Wire to Y on thermostat. (Compressor contactor low voltage)
The (G) note the color of this wire and trace this back to the thermostat. Place this on G on the thermostat (Fan Relay)
The (W) terminial, again trace this wire to the thermostat (heat)
The (R) terminial, depending what type of thermostat you have , if you have Rh & Rc terminials on the new stat, you may have to place a jumper wire between these to terminials on the thermostat. Is there two wires on the R terminial? If so place one on Rh and the other on Rc, on the thermostat. Does the outside airconditioner have its own 24 volt transformer? Then one of the wires on the transformer (r) will have to be placed on the Rc terminial of the thermostat. This is very rare. There would be more that 2 low voltage wires going from the furnace to the outside AC unit being used if this was the case.
If you have trouble figuiring out the low voltage wires. Turn off the power to the furnace, note location of the wires on the terminial board, write down the color or use colored tape (red , white, green, yellow, etc) to correspond with the terminials. Use a Ohm Meter and trace the wires by touching two together at a time at the thermostat location. As you find them R,W,Y, & G mark them with your colored tape.
If this is "just to much" then you will have to hire someone "in the know." Hope I was of some help.