@Christian, very simple hookup, even for the novice. You have done half of the work already by taking the photos of your current wiring. From the pictures it looks to be that you have 5 zones, Dark Blue wire Zone #1, Yellow wire Zone #2, Light Blue wire Zone #3, Orange Wire Zone #4, Black wire Zone #5, and the White wire Common.
It's pretty straight forward, remove your wires from all zones on your old controller box. Don't worry about the power side you won't be needing those wires to be unhooked as you won't be needing that AC adapter. Feed the Zone wires out through your old controller box once unhooked, unscrew/mount the old controller box from the wall. Mount the Rachio box on the wall, feed the Zone wires as per above Zones #1 thru #5 and attach each Zone in each slot, attach your Common wire to one of the Common slots marked C. Plug in the AC adapter (I pre-downloaded the App) and configure your wifi for the Rachio controller to go live. Then configure your 5 zones, naming your zones, picking your type of grass, soil, exposure, nozzle, slope, etc. Then all you need to do is configure a watering schedule.
That's pretty much it, you can alter each zone, changing watering frequency later as you become more familiar with your Rachio if your not sure of the advance stuff. My advice is to read the many post on this Community site to get better familiar with your Rachio and post questions or problems you may be experiencing. Also they have a great Support staff that can help you with any problem you may have, or questions you need answered.
That's pretty much it.