@Thejokerabides - From the schematic at the bottom, the white wire is the common going to the solenoids and will go in one of the Rachio C terminals. The black wires are the zone wires, put each of them in their own individual terminal. The yellow, red and blue wires are internal to the Rain Bird transformer and can be ignored - you won’t be able to connect to them when the old controller is removed as they’ll go with it. The equivalent for the Rachio will come from the power adapter plug.
Before you disconnect the Rain Bird, run a quick test to make sure all the zones are working. Then label the wires, remove and replace.
If you see a rain sensor in the installation, it is probably in-line with the common wire out to the solenoids. That will work, but can lead to questions about why didn’t my system water when the interface says it did (i.e. the rain sensor opened the common wire).