@JKirkChavez - the big white wires on the AC do not get used in the Rachio. Not knowning what else is installed (rain sensor?) I can’t comment on the smaller wires going into the AC terminals. For the zone wires, I’d put both in the Rachio zone terminals.
Run a quick test using the Irritrol to make sure all the zones work before migrating to the Rachio. This will help in troubleshooting if a zone doesn’t work.
Welcome to the community.