Drew. Thanks for reaching out to us. Sorry for any confusion with your wiring. So your main wires are the white common wire and the 4 zone wires below that. With our controller simply connect the white common to the C slot and then the corresponding zones 1-4 on our controller as well.
The upper left section are your 24VAC and Ground slots. You will not need to put these wires into our controller since you will be using the 24V Power Adapter that we provided to power the Rachio. The Red and Black wires are for a rain sensor, which I assume you have. If you want to connect those to the Rachio controller you can, but a rain sensor is not necessary with our intelligence. That being said if you want to connect your rain sensor you can. I am not positive which Rachio controller you have, our Gen 1 or Gen 2, so I'll explain how to connect to both.
Gen 1 is pretty simple, just connect the Red and Black wires into the yellow R1 and R2 slots. It doesn't matter which color goes where. If you have a Gen 2 controller you will put the Red and Black wires into the S1 and SC slots which are positioned in the middle between the zone wires. This article shows how to do that. Again the rain sensor is not necessary so if you don't want to connect them that is fine.
That should do the trick. Let me know if you have any other questions Drew.