After checking the battery and battery connections, try to crank it again. If still nothing, take a look at the solenoid. There well be two large wires to the solenoid. One goes to the battery, the other to the starter motor.You listed a symptom of the starter solenoid clicking. Thomas gave you the first step in troubleshooting. To that I would add have the battery load tested. It is usually less expensive to begin with a systematic approach to troubleshooting than to start by disassembling things.
And if the starter doesn't spin, then most likely the starter needs rebuilt, as in cleaning, brushes and bearings.If this test causes the starter motor to spin, then you can rebuild/replace the solenoid.
Yes, that's what I'd do. Many times OEM starters are much better after a rebuild than replacing the original with a box store aftermarket starter.And if the starter doesn't spin, then most likely the starter needs rebuilt, as in cleaning, brushes and bearings.
SR