Do the voltage checks FIRST, undervoltage from a bad cable, bad connection or a poor connection thru a solenoid can lead to a LOT of other problems. Volts go down AMPS Ramp UP to compensate to make the same level of power, AMPS are HEAT.
Voltmeter, Check POS Cable V at battery to Grd, last connection to go out to starter then check at hot post On Starter, needs to be Within .5v of Battery v at Starter. Next up one multimeter lead to POS cable at Batt other multimeter cable to Hot Post on starter, need to see that as a NO CHANGE "0" as if shows a voltage there is a problem Same for Negative Cable, should show NO voltage. Voltmeter check with someone getting the 'Click', first at Battery P-N should see Battery V if not feeding forward, SAME at Starter Hot Post to GRD If goes hard LOW V at Battery then the starter could be locked, If Remains at Battery at Both ends then find the 'Click' mechanism, if is a secondary solenoid will have Two Large Terminals, One Hot the other Not until engages, then BOTH should go hot, the other end of that output cable will be the small terminal at the Starter Motor Solenoid, it had better show hot of the at small relay shows hot or wire is broken corroded in two, damaged. The other possible is a NEUTRAL Lock Out Switch much as a clutch switch, you will need a schematic to find it.
Will need TWO people to do most of these tests, one at Multimeter one on switch.