The right way to do the job is with an industrial rated Programmable Logic Controller (PLC). All inputs and outputs will be appropriate for the application and since outputs are usually relays (variations do exist), driving the solenoids directly is really easy. Yes, this is the way to do the job.
OTOH, once you know the states (read up on state machines and C switch() statements) required for the operation (and you seem to have a start above but consider how to handle possible invalid situations), you can code that transition table in a C switch() statement. Arduino provides delay functions if these help.
HOWEVER:
There are hundreds of PLCs, I searched for a 24VAC PLC and came up with this at Newark
https://www.newark.com/crouzet-automation/88974044/controller-plc-8-i-p-4-o-p-24vac/dp/64R1068The programming software is cheap (in comparison)
https://www.newark.com/crouzet-automation/88970111/millenium-3-software-cd-rom/dp/76M8857?MER=sy-me-pd-mi-acceThere is a cable to connect the PC is pricey but not outrageous
https://www.newark.com/crouzet-automation/88970109/usb-programming-cable/dp/86K0477?MER=sy-me-pd-mi-acceHere's the thing, learning C or PLC programming takes about the same amount of time so it's a toss-up. The difference will come up when spending hours to design the hardware and get a PCB laid out. My time is $100/hour (not that anybody paid me that) so it doesn't take much time designing custom stuff before it is just better to buy fully qualified equipment and be done with it.
Yes, the Arduino will be a learning experience and is well worth the time/expense. Actually applying it is not likely to be as successful as you might hope. I would definitely go with the PLC and learn to program in both environments.
Using ladder logic as your PLC design scheme means the system is almost portable to ANY PLC. I would say EVERY but, somewhere, there will be a PLC that doesn't take ladder logic. I have never seen one, I have never even read about one, but that doesn't mean one doesn't exist. The thing about ladder logic is that it is as portable as it gets.