Hello, its late I have been up for over 18 hours and am going to bed shortly.
Assuming that the system was wired using the commonly accepted standards, you can connect Red wire to R, Green wire to G, and Yellow wire to Y, and white wire to W.... that leaves us one mystery wire, that would be the one we connect to C
We identify that C wire using a volt meter, and put one test probe on Red wire, then try other probe on any of the remaining wires that have not been connected yet... the wire that reads 24 volts is the one we connect to C.
Assuming there are no defects in the system, and that the wiring in the furnace/air handler is correct, and that your thermostat is OK.... you AC should run.
I will wait up for another 20 minutes tonight to see how it goes for you.... please do not forget to rate my work positively if it runs, lacking that a lot more trouble shooting will be required and we can do that after I get 6 hours sleep tonight.