Matt D,
It could be your solenoid but it more likely is a bad wire connection or a bad starter/ignition switch. Refer to the "Wire Diagrams" link via the link on the top home page. I like Roland's Fancy Color diagrams the best.
I'd start with a look at the ignition switch first. Often the terminals on the switch rust and corrode over years of service. Pull the 3-spade wire terminal off the back of the switch. Check the three spades on the switch to see if they are rusted and/or loose. I've had one or more of those spade terminals actually pull off by just pulling the connector off due to rust/corrosion which would be the key indicator to your no starter problem and switch needs replaced. If the terminals are tight, clean the terminals and reinstall the connector and try to start.
If no go, next check the solenoid by connecting a jumper wire to small bolt terminal with an orange/black striped wire running to it from the starter/ignition switch. Then momentarily contact the other end of the jumper to the battery positive terminal (Note, this will cause some small sparks). If the solenoid clicks and the S/G spins over, then your solenoid is okay. If it just clicks with no go, then remove and clean the larger primary terminals and red cables on each side of the solenoid. Reinstall and try again. If still no go, then your solenoid is possibly bad. I say possibly bad because you could have a poor ground.
Check the ground wire from the battery to the frame. Clean the ground wire terminal where it attaches to the frame as well as local area where it attaches to the frame to remove any dirt/rust. Also carefully inspect each terminal end of your battery cables for corrosion in the wire itself. If this is the case you will see green/white powdery stuff where the terminal is attached. Often the insulation is frayed or bubbling up and you not getting good continuity through your cable due to the corrosion.
Try to start again. If no go, then try a different solenoid. If you have a continuity tester light, you can hold the probe end to the terminal with the red cable the runs to the S/G and if it lights up when you hit the starter switch, that would indicate that solenoid is working.
Still no go, then focus on the S/G. Clean and check the wire connection at the armature terminal.