Air moves but is not cold. The fan is fine and the refrigeration side is not doing its job. Common causes are a failed capacitor stopping the outdoor fan or compressor, a contactor that is not pulling in, or a refrigerant charge that has leaked down. Refrigerant is not consumed in normal running, so low charge means a leak, and topping it up without finding the leak just sells you the same repair again next summer.
Barely any air moves. The problem is airflow. A blocked filter, a failing blower motor, a frozen coil or a disconnected duct all show this way. If the coil is frozen, the airflow restriction usually caused the freeze rather than the other way round.
It cools, then stops, then starts again quickly. That is short cycling, and it has several causes including an oversized system, a failing control, restricted airflow or a refrigerant problem. It is hard on the compressor and worth diagnosing rather than living with.
Past that point, judgement by feel stops working. Whether a charge is correct is a measurement of pressures and temperatures against the manufacturer's chart for that model, in the conditions on the day. A capacitor that looks fine can be well out of tolerance on a meter. A compressor that runs can still be failing electrically.
Our diagnostic visit is a paid visit and we say so before we come. You get the readings and a written price before any repair starts, and if the honest answer is that the system is near the end, you get told that with the numbers behind it rather than a quote for a new one.