If the indoor fan doesn't turn on, (1)check to see if the outdoor condenser is working, if so you likely have a bad indoor motor, (2) if neither turn on check your breakers, if their not tripped. (3) you may have an internal HVAC electrical short.
(1) Have you changed the air filter? (2) Have the refrigerant levels been tested? (3) Is the compressor running? (4) Are the coils clean indoor and outdoor?
(1) Have you changed the battery? (2) Have you checked your breakers? (3) have you recently replaced your thermostat? Normally the there is a problem with power being supplied to your indoor furnace or air handler.
(1) Is your condenser more than 10 years old? (2) Has maintenance been performed? (3) Is the fan spinning while in operation? Typical cause is a compressor that is on it's last leg or low on refrigerant.
(1) There is only one reason for this one, the indoor motor is failing, turn off your HVAC unit immediately and get service soon. This is a certain fire hazard.
(1) Has maintenance been performed? (2) Have you installed Carbon-Monoxide detectors? (3) Are your door panels safely secured to your heating unit? Carbon is not detectable by without detectors and can be deadly. Be safe and schedule maintenance.
Just like the A/C portion, your system must be able to breathe. (1) Have you changed the filter? (2) Is the Evap. coil clean (3) Is the indoor fan running? The usual problem, is trapped heat. but it can also be safety sensor detection.
If your indoor fan never turns off, this can only be an issue with a failed fan relay.
Your HVAC system is likely a heat pump, when fan relays or control boards fail. The heat pump condenser will miss it's defrost cycle.
Having high ceilings can almost always remove any efficiency from your air conditioner or heater. It takes a considerable amount of energy to fill a room with cool or hot air. There are some great tactics to try such as increasing the amount of insulation in your attic or try tinting the windows the sun hots most