AC not working on a 2005 Chevy Sonic

I took my 2015 Chevy Sonic to the dealer because my air conditioner wasn’t working. I’ve had a million problems with this car but it’s been under warranty till now. Theyve even replaced the motor. Or so they say. But they told me it was the compressor and it would cost 600$ and they didn’t even know if that would fix it. I’ve been looking online at some things but was just wondering if the fan kicks on when I turn the air conditioner on does that mean the compressor is working? Is there a way to tell if it’s working or not? Also it is still slowly leaking antifreeze and has overheated several times.