The click from the starter is the solenoid engaging, so you're good there. This narrows it down to two things, either its not getting enough power through the main cable to turn over, or the starter is locked up. If you've already checked connections, I'd have the starter tested.
If you want to test the starter yourself, you can use a jumper cable to connect the battery to the starter terminal. WARNING: its really easy to short the jumpers to something.