Bert is correct as far as the circuits go. If you can jump the starter and it works, then that end of the circuit is good.
on your ignition switch, there is a B and S terminal. The B terminal will be hot all of the time with battery voltage. when the key is turned to the start position, the S terminal will be hot. this will be either an orange or red wire that will go through the PTO (deck) switch and the brake switch then to the solenoid. If you turn the key to start and check the small red wire at the solenoid, it should be hot. if it is not, it is either a wire or safety switch. If it is hot but not energizing the solenoid, then it is either a bad ground to the solenoid or a bad solenoid. run the mounting bolts in and out a couple of times to clean up the threads. if the solenoid activates but the starter still does not engage, it is a ground issue on the engine. Run a wire (12ga minimum) from the engine to the frame. It should work at this point.