Everything turns on indoors except the outdoor condenser. It is only when I push down on the contactor that the outside unit turns on and cools perfectly, but it doesn't stay on. I suspect it's my relay or capacitor? Any suggestions? The relay part # is ICM205 it runs about $16 should I change this?

If 24 Volts is present on contactor coil(low voltage connection) and it doesn't pull in, It is Bad.
If 24 Volts is Not present(when set to cool), it could be thermostat or wiring.

It takes 24 volts to close the contactor and only one open safety switch or time delay to keep it from closing. If you can give a make and model of the system it will help us to get you started down the right path of helping trouble shoot your problem.