How old is the battery? I've found that they typically don't last more than about 3 years on these little tractors. Take to the parts house as the OP suggested, and a load test will tell you real quick if you have shorted or open circuit plates. But, before you do that, clip a jumper cable on the terminal of the starter, making sure it doesn't touch anything else. Touch the other end of the jumper to the POSITIVE (+) post of the battery. If it turns then, the battery is fine. You're gonna need a solenoid. A "Starter Relay" for an early 90's Ford pickup serves well, and doesn't cost nearly as much as the OE solenoid. Be very careful not to short the battery out by touching ground (anywhere except the battery + post or the starter stud), or you may indeed wipe out your battery, if you don't blow it up. The click with no attempt by the starter to turn the engine over suggests either a bad solenoid or a bad starter, if not something so simple as a loose connection in the starting circuits (big wire connections). I've also had one instance where the oil film that builds up between the frame and the engine caused a high resistance ground in the starter circuit. A little corrosion (if you regularly use this for snow removal) goes a long way toward causing problems.