Hi, and welcome to the club
Looks like you will have a pretty straight forward time to switch over your old system to new.
Step 1: Connect the 5 thick white wires to terminals 1 through 5 of your new controller. Order does not really matter as you will be able to nickname the zones and take zone pictures during the setup.
Step 2: Connect the thick red wire to any of the C terminals of your new controller, to do this you will need to disconnect it from the thin white wire that it is currently connected to.
As @DLane has correctly pointed out, here are optional steps, do these only if you wish to use the rain sensor you’ve got.
Step 3: Take the thin white wire that used to connect to the thick red wire in Step 2 and twist connect it with one of the thin yellow wires (which one does not matter) that is coming out of the same black cable. You will wish for these two wires to essentially become one.
Step 4: Connect the above mentioned wire pair (white and one of the yellow) to the SC terminal (in case you have Gen 2) or 24 V- terminal (in case you have Gen 3).
Step 5: Connect the other yellow wire that you’ve not used for step 3 to SP terminal (gen 2) or 24 v+ terminal (gen 3).
Step 6: Connect the blue cable to the S1 terminal and turn on the rain sensor within the Rachio app during the setup.
P.S. Don’t hesitate to ask questions, and post the photos of your completed setup