If this thing is indeed a TRX300, then yes it does have an electric start. With the key on, in nuetral, when you push the start button, what happens? Does the engine turn over or do you just hear the solenoid click? If you just hear a click, either the solenoid is bad or the starter needs rebuilding. If you have a test light or volt meter, put it on the cable where it attaches to the starter motor. If there is voltage, the starter is the problem. If no voltage, then its the solenoid.

As far as the kick starter, no you don't have to split the cases like JimmyHoffa says (No wonder they can't find your body, here you are on the internet all this time!!). Remove the right engine cover to access the clutches. Your problem more than likely will be the one-way sprag clutch thats on the centrifigal clutch. Use your manual to help you diagnose and repair.

 

Good, I didn't feel like splitting the whole thing. I guess I'll have to wait for the manual anyway. And unfourtunately the problem wasn't my solenoid. I took the stater apart and its so far gone I don't even think it can be rebuilt, no brushes, springs, rust everywhere. So I think I'll just be buying a new one.
Thanks for the info.