I like to start from the starter motor and go backwards .
Do the following 5 tests, regardless of the results from an or all of them as there can be more than one problem and you want to isolate where the problem lies.
Elimination of individual parts is important so you know by the end, the battery, solenoid & heavy power circuits are all in good order.
1) try to jump the starter motor directly from your car or truck.
Starter turns = starter good
2) do the same directly from the mowers battery
Starter turns = mower battery good
No turn = duff battery, recharge it & try again.
The engine turns over and starts.
3) check for voltage ( + 12V ) at the solenoid trigger wire with the key in start position
3a) same with ground trigger wire ( 4 wire solenoid ) or body of solenoid ( 3 wire solenoid)
( I like to test V from the battery hot terminal to ground terminal rather than ohms as they give funny readings )
Will test. The solenoid is a 4 wire. Two main top post and two smaller wires on the side, one white, the other an unknown color.
4) leave ground jumper in place ( from step 2 ) & try key start.
Starter turns = power connection good but ground connection suspect ( most common )
Confirm it by trying again, extra ground removed
I run a secondary ground from the grounding bolt to one of the starter mounting bolts & paint over both with liquid electrical tape.
What ground wire from 2) ?
5) Remove the trigger ( thin ) wire / wires from the solenoid.
Ground one & bridge from the hot terminal to the other.
Starter cranks = solenoid good.
Solenoid is not polarity sensitive, BUT THE WIRING IS so make sure you remove the thin control wires.
Note a thinner wire on the hot terminal is not a control wire. It is the main power feed to the mower.
trigger wires on this solenoid are the two smaller wires right? 
From here on things become very mower dependant as starting circuits are getting changed all the time.
Basically the power goes in a loop from the hot side of the solenoid ( saves wire, no other reason ) through the fuse to the B terminal on the key switch then to the PTO switch then to the parking brake switch then to the solenoid trigger switch , easy peasy after you grow the 3rd arm. Use a test lamp and follow the power.
However a lot of mowers with a 4 pole solenoid, run a secondary ground control circuit to the ground solenoid wire through the lap bars.
Then to stop this interfearing with the normal safety function of the ground kill, it goes to a relay with the ground as the switched connection.
These are a PIA as the + control wire to the relay comes from the power loop above and the ground side of the control comes via the normal cut out functions of the lap bars.
Be very careful because if you have a system like this and accidentally send 12V down the ground loop you can fry the magnetos on some circuits.
