I swapped out my homebuilder-provided Rainbird ESP-ME3 controller for a Rachio 3 last year and it was straightforward. Thomas_Lerman provided a very good, detailed list of what to do. Seeing that many of your zones have red wires, labeling them before disconnecting is important!
For the sensor that is connected to your Rainbird, if it is a rain sensor, I suggest you check out this page before re-wiring: Rain Sensors FAQ
I have a rain sensor and eventually disabled it within the app, since activation notifications were a nuisance and the Rachio does such a good job of integrating local weather information (I am using the Flex Daily schedule and am connected to a local personal weather station).
I hope this helps, and best of luck!