The color codes for each wire is in the full manual w/ schematics up on IPDB (if you don't already have a copy).
Bottom middle of the "Plug Connectors" page has the Wire Color Code key.
So example, J1, pin 1 shows as color "50", which translates to 'white, no trace'.
Or J3, pin 5 - color 92 = 'grey with blue trace'.

The transformer/rectifier board is the easiest to do first, because its just a matter of testing the voltages at the TP's, and then resoldering/re-pinning the connectors.
From your pictures, my main concern is the battery damage on that MPU board.
Most repair information can be found on the https://www.pinwiki.com/ early Bally section. (including how to build a bench tester, although I've personally never done it... I have several Ballys, so i tend to just swap boards into working machines to test.)