Some friend gave me his White (MTD) riding lawn mower with all the wires removed and hanging from a three point solenoid. There is a black thick wire that returns to the negative side of the battery- should it be just grounded to the chassis and not to the solenoid? I think I can figure out the positive battery cable goes to one large solenoid post and the starter power cable to the other post.

When the ignition switch turns into the "start" position that is what activates the solenoid. That is the wire that should eventually run to the post on the solenoid. The reason for saying eventually is because on some models that wire was diverted to safety switchs. So depending on age of equiptment. It may or may not run directly to the solenoid. If the unit uses body ground (negative) then the starter needs positive power to function. So positive battery cable runs to one side of the solenoid and the other runs to the starter. When the solenoid is turned on it allows the positive to connect to the starter and it should start turning over (cranking).