The most common thing to cause the A/C system blow warm or cool-not cold air is that the freon charge is low.
Compressed refrigerant from the compressor enters the condenser in a high temperature, high pressure vapor state. As the refrigerant flows through the condenser, the heat of the refrigerant is transferred to the ambient air passing through the condenser.
Cooling the refrigerant causes the refrigerant to condense and change from a vapor to a liquid state.
The condenser is located in front of the radiator for maximum heat transfer. The condenser is made of aluminum tubing and aluminum cooling fins, which allows rapid heat transfer for the refrigerant. The semi-cooled liquid refrigerant exits the condenser and flows through the liquid line to the orifice tube. The evaporator gets extremely cold. Air from the blower motor passes through the evaporator core, is cooled, then enters the passenger cabin.
The first step in diagnosing poor cooling is to check that the compressor is engaging when you turn the system on. If it does not, first check all fuses. Next the freon charge needs to be checked. just adding a little from a can you purchase at a parts store, and assume you have enough in the system, is not a good way to start. Without gauges, you have no way to know how much was in prior to adding. If the system has a gib leak, there will also be air in the lines, so if pressing the small pin the service port releases pressure, it may just be air, and not freon. The best way to check is to have freon drained and measured. then the proper amount can be re-charged.
If the compressor still will not come on, then it is time to look at a possible electrical problem. Using a test light, volt meter and schematics, as well as a scanner, are tools required on today's cars and trucks.
Once the compressor runs, the pressures will need to be checked. If pressures are good, then looking at the air delivery system in the dash is next.