Some depends on the genset, but I would start with :
Old gen start 2 and gen stop 2 (both yellow) to new #16 and #18 - note both should be ground, so it shouldn't matter which is which, and IIRC, I hook all of those together *and* to new #13, which is ground.
Old Thermostat + to new #1 - Thermostat 1 input.
Old Gen start 1 (black) to new #17 - gen start, normally open
Old Gen Stop 1 (blue) to new #16 - gen stop, normally open.
Old Gen run (brown) to new #12.
If it's an Onan, Type 2 should be the ticket, maybe type 1.
-- Chris Bryant