Sounds like you're on the right track. I'd start with the neutral safety switch. Hold the key in the start position and move the shift lever back and forth. Sometimes the switch can just be a little out of adjustment. No luck there? move on to the starter and solenoid.
First things first, make sure all the power and ground connections at the battery, solenoid, and starter are clean and tight.
To check the starter, use a screwdriver and short across the two big posts on the solenoid. BE CAREFUL when you do this. It will likely spark pretty big and the engine should turn over. If it doesn't, you've got a problem with the starter most likely cause by being submerged by the water in your bilge. If it does turn over, the starter is fine, and you'll want to start checking the solenoid.
First, check if the solenoid is getting voltage from the ignition switch. If it clicks when you turn the key, its getting voltage. If it doesnt click, check to see if there is voltage between the two small posts on the solenoid while a helper is holding the key in the start position. If its getting power and doesnt click, the solenoid is shot. If it isnt, start tracing the wires from the solenoid to find where the fault is.