Galaga1. Please don't private message me questions. Not that I mind it's just so that we keep everything above the board so that other can reference these threads and possibly learn from our conversations/suggestions.
With that being said, from what you are telling me, the fan works when you manually engage the contactor. However, you didn't perform any of the troubleshooting tests that I suggested.
You will know when the compressor comes on when manually engaging the contactor because you will hear/feel it working. It will start out loud and then smooth out within seconds of working. It's hard to explain but, there is a difference between the sound of a working compressor and a locked up compressor. It is possible that the new install of your furnace is the cause of the problem from an electrical standpoint. It sounds like they did not hook up the 24VAC wires from the furnance to the outdoor coils on the contactor. Performing a volt test on the coil wires while in A/C mode will reveal voltage or you can visually inspect the T-stat terminal strip in the new furnace and see if they hooked up the wires and/or hooked them up to the correct terminals.
Post back here if you need more clarification and/or help.
I'm more than glad to guide you through the steps but, please keep your posts where everyone can see them and help me help you. It doesn't do any good if they are not part of the conversation. Also, it helps me if you follow through with the tips I provide so that I can narrow down the problem more quickly.
Welcome to the forum, we are more than happy to help.