If you want good information form this forum, you have to feed it good information.
Like the model number of your mower off the ID tag and the make & model number of your engine
Thusly armed we can acquire a wiring diagram to ensure the information we give you is correct.
The solenoid will have 1 or 2 small wires at it's base.
These are the trigger wires.
If there are 2 one will be + & the other - when the solenoid closes to energise the starter motor
One might be switched or both might be switched.
There are subtle differences but basically all work the same.
A power feed wire come off the hot terminal on the top.
This goes to a fuse then to the various switches & back to the + trigger at the bottom
Circuit is
Fuse to B on ignition switch
B to S on ignition switch
S to PTO ( safety switch or electric PTO switch )
PTO to brake / clutch switch
clutch switch to solenoid.
This is a 12V + circuit so you can test it with test lamp hooked to a ground, or a mulitmeter measuring volts.
Back probed by stuffing the electrode up the back of the plug.
It will be helpful to grow a 3rd & 4th hand or make a short jumper with a spade at each end.
Once you have confirmed that the ignition switch is working, pull the plug off the back and jump the plug which will save you having to turn the key to S every time
It is prudent to remove the blower housing and remove the kill wire from the magneto because if you accidentally send 12 V down that wire for the briefest period you have fried the chip in there
Most safety switches are double switches which have a power side and a ground side
When looking at the switch the pairs are flat to flat not edge to edge.
Basic wiring information can be found in a Briggs book called the repower guide which should be in every ride on owners shed
https://www.vanguardengines.com/content/dam/briggsandstratton/na/en_us/Files/FAQs/alternator_replacement.pdf