I helped a friend with her Ducts this past summer.

Cats in the crawl space, need I say more.

The AC worked fine but after a while the on/off function started to act wonky so I replaced the wall controller in the hallway. I truly believe the controller was bad because it didn't respond when the buttons were pushed. Pretty much like a dead battery even with new batteries.

New (very basic) controller installed and the AC worked fine for the rest of the summer.

Switch to heat and everything worked fine for a week or two, then the heat wouldn't turn off. They did something to the controller until it went off. Then it wouldn't turn on when the temp was well past the cut on point. Auto/On - Fan switch works ok.

It appears to start and stop for no reason and it will heat the house to the high 70's when the controller is set to 65 and then shut off for no telling how long before it starts up again.

The other night they asked me where the circuit breaker is to shut it off. I told them not to because the heater has to cool down after the gas burners shut off and killing the power will stop the fan from cooling it down.

I understand AC/Heat to the point of cleaning and replacing parts but I have no idea where to start on a system flaking out. I told them call a full time AC/Heat guy but it bugs me to not know what is happening.

Any thoughts?

Hard to tell with limited info here.
How old is the furnace? What make and model?. What thermostat are you using and is wired correctly?
Typically a forced air furnace with A/C will have a 4 wire T-stat.
The wires will usually be:
- Red, which is the 24vac supply from the furnace transformer up to the T-stat
- Yellow, which is for a call for cooling by the T-stat
- White, which is a call for heat by the T-stat
- Green, which turns the fan on manually from the T-stat.
If you have a 5th wire which is sometimes black that that is most likely the common wire from the transformer.
That is used as the other leg to power the T-stat.

So if you remove the t-stat and connect the the red and the white wires together. The furnace should start the heat sequence.
If you connect the red wire to the yellow wire. The A/C should start, but don't do this if the outside temp is below 50 degs.
You already said that the blower starts when you switch it to "on" at the T-stat so you would not need to do the red wire connected to the green wire.
The T-stat is nothing more that a temperature switch that sends the red wire's 24vac back down to the furnace on one of the other 3 remaining wires
to call for heat or cool or manual fan operation.
Also, I'm assuming that this is not a heat pump...