Switch from old mechanical hand-cranked timer system?

I did a quick search for

transformer “S84A-410”

and the first page shows all 24VAC. It does not hurt to double check, but I would guess you are good.

Just for my own sake, I will transcribe what I see in the wiring picture:

Transformer:

  • White goes to valves’ white, which is Common (‘C’ on Rachio)
  • Red goes to the timer’s black with the opposite end of the timer going to one leg of both switches

Valve wires

  • White already mentioned
  • Red & blue are the connected value wires: one would go to ‘1’ and the other would go to ‘2’ on Rachio
  • Yellow & green are not connected and appear to be unused zone wires

The only other thing left is power to the Rachio. Hopefully you have an outlet nearby which you can plug the Rachio supplied power supply into and connect the barrel connector on the other end into the Rachio. Do not connect any other power wires to the Rachio. In the worst cast, you or an electrician should be able to replace the transformer with an outlet.