You start by noting the model & serial numbers off the ID tag .
Write them in the space provided on your owners manual ( no one does ) so they will be handy.
Next go to the scag web site then download the parts list.
From memory the wiring diagram will be at the back of the parts book.
Post the wiring diagram , or the whole book if you don't know how to pull a single page out of a pdf.
The test to see if the alternator is charging the battery is to check the voltage across the battery
1) ignition off
2) ignition on
3) engine on and running full speed
4) engine on running full speed, blades running.
That will confirm what we currently suspect
The next step is to work out why and for that we need the wiring schematic so we can all be singing from the same song sheet.