You can visually verify if the compressor is running by watching the clutch engage and disengage with the climate controls set to A/C. If it's engaging and immediately dis-engaging on a warm day, the system is probably low on freon. If it's not coming on at all, you'll want to check the relay and the wiring. The relay is in the box on the driver's fender side under the hood. You can swap in a known good one or have a finger on it while somebody toggles the controls in the cabin, you should be able to feel it engage. A multimeter on the terminals works, too, but you'll need to know which ones to check. A service manual would be handy for looking that up. If the compressor is getting power but it's not coming on, the system could be totally empty or the pressure switch might be faulty.
Check the condenser up front with the radiator (trace the lines from the compressor). Is it dirty?
If the compressor is running, but you're not getting cold air, you might have a clogged expansion valve / orifice tube. It's up under the power steering resevoir.
If it's hot when stopped and idling or moving at low speeds, but fine when at highway speeds, you might have a fan problem (or a really dirty condenser or a lot of bent fins on the condenser, or some combination of both). When you turn the system on, one fan should engage immediately and stay on. The other will come on when the powertrain controller decides things are getting too toasty under the hood. You can verify operation of both fans by pulling the coolant temp sensor connector. It's on the driver side of the engine, at the front, in a plastic housing. With it disconnected, the powertrain controller should command both fans on to maximum since it doesn't know what the coolant temp is. Watch out, this could throw a code and illuminate your check engine light.
If the compressor is staying on for a while, check the pressure in the system at the service ports (stay away from the high pressure port unless you really know what you're doing) or have somebody else do it if you don't have a decent set of guages. If there is too much or too little freon, the system won't cool.
I've been in this type of vehicle in the summer. You just don't seem to get really cold air out of the vents when it's really hot outside at low speeds. Though I've never been in one that was factory fresh, so I can't swear there isn't something wrong somewhere, but I've checked everything I mentioned. Outside of stop and go traffic, 40 degrees at the vents is doable on Max A/C once the cabin has had time to cool down.
Oh, and a pressure test after 1 day does not prove the system is tight. Just that if there are any leaks, they're fairly small. If it's still at spec pressure after a week or more of running in hot weather, it's tight.