[Thumper]

Shouldn't that loose red wire go to the battery? The two main poles on the solenoid are connected in the solenoid when you press the starter. This connects the battery directly to the starter. The battery ground is connected to the engine case to complete the circuit (this connection would be a good one to check-possible bad ground).

You can test the starter, and the solenoid independently.

1. Test the starter (bypass the starter button). The solenoid completes the circuit from (connects) the battery to the starter when you press the starter button.

That is, it connects the thick wire from the battery to the thick wire that goes to the starter. These are connected to the solenoid main posts. So you can simply connect the two by using a heavy copper wire and touch (connect) these two terminals on the solenoid (or use a screwdriver!). Make sure transmission is in neutral first.

If there is enough juice in the battery, the starter will run...turn over the engine when you bypass the solenoid. No need to have the key turned on (or the engine might start!).

2. Test the solenoid. The starter button may provide 12V to the solenoid or it grounds the solenoid. You can test for either of these. Press the starter button with a charged battery in place. Use an ohmeter to see if pressing the starter button gives 12V to the solenoid. Then test if the solenoid is grounded when you press the starter button.

You mentioned that your solenoid clicks when you press the button, so presumably that starter circuit is intact.

Either way, connect an ohmeter to the two poles that have the heavy wires (one to the battery, the other to the starter). Press the starter button. When you do, the solenoid should click and the posts with the two heavy wires will be connnected if the solenoid works. Ohmeter goes to zero ohms if the solenoid works (connects the two posts...)

If the starter and the solenoid both work, you have a wiring problem, probably a weak ground since it does click when you press the starter button.