Aruuuuuuuuuuuuuug. You may need all the help I can give you. Well a good tech might be even better. This is how things work:
Usually there is a 24 volt AC transformer in the furnace with the secondary winding connected to a red wire running to the thermostat and a blue wire, common, to the gas valve, A/C relay, and fan relay. From the thermostat there will be white wire to the gas valve, yellow to the A/C, and green to the fan. The thermostat is wired to switch the power from the red to the white, yellow, and green as needed with the blue completing the circuit. Most thermostats and furnaces have the contacts labeled R, B or C, W, Y, and G for the corresponding wire colors. It may be wired to have the A/C control wires return to the furnace and its controls and then a second wire goes to the A/C unit. Internal wiring may replace the green wire if the thermostat does not give you the option of fan only or continuous fan. Digital or programmable thermostats may need the blue wire connected to them.
I came across the niftiest gadget for trouble shooting, a voltage detector. There are several brands. I have a GB Instruments GVD-505A, less than $15 at Home Depot. Touch it to a hot wire, and the end glows red. Find the doodad that lights it on one side, and not the other, and you have the culprit.
Cut wires are where cut wires are. I had control trouble with my old furnace and A/C several times. Eventually when I tore it out to put in a new one, I found an inaccessible place where the chipmunks had eaten the insulation off the thermostat wires. As above, you must have a small pair of wires going from the furnace to the outside A/C unit. Usually they are run out of sight if possible. If something happens to one in some hidden area, running a new one down the side of the house might be a temporary solution. Since this is a new installation, I hope you won't have to pay most of the expense. If you drove a nail in the wall the wrong place or something, likely you will have to pay for that.
Here is my new A/C unit with the armored wiring and insulation.
One more point. Do you have a little pipe coming out of the attic that drips water when the A/C runs? Track it down and keep an eye on it. That is the moisture the A/C takes out of the air. If it quits dripping water when the A/C runs, that means the little drain is plugged up and soon water will be running down on your ceiling. You might ask about a float switch and an alarm.