Pictures were a bit small, but showed enough to see what matters
You can disregard most of the wires connecting to the left hand side, those include power (pair of yellows & green) and remote (Red, white and blue). The two remaining baige wires are your rain sensor and can be connected to S1 and 24V- terminals of your new Rachio Gen 3 controller (more on this here), they are not strictly needed, but you can hook them up and than choose to use them if you wish (via setting in the app).
The wires on the right (White through Brown) are your zone wires, so simply disconnect them from the old controller and hook them up to the new. White wire goes into any of your two C terminals. The rest would be hooked up to the zones (1 - 8). Order does not really matter as you can name the zones and take pictures during the setup, irrigation by number if a thing of the past.
You do not have a Master valve wire hooked up to your old system. So you can simply choose default options when software is asking you about it.
Congrats on the upgrade and welcome to the community