Hard Wire

I am replacing my old system. The old system is hard wired under VAC. I noticed the Rachio 3 device has input for 24 VAC + - can I just connect the same wires there or do I need to connect and use the power adapter supplied?

Rachio does not have a place to run hard wires to. The only place is the barrel connector without opening the controller and soldering which I do not recommend. If you absolutely cannot use the supplied power supply and know what you are doing, you might be able to connect the old wires to a new connector. If it were me, I might be nervous about messing it up and causing issues.

No, even thought it may seem like they are the same, 24v terminals are fed through a small filter and not protected by the input fuse or surge protector.
Running the controller via terminals may work for a short time, but eventually those terminals or the controller itself will fail.

I recommend you keep the original supply as test / reference unit. And get a short 5.5x2.1 barrel cable to terminate your old wires. I recommend you avoid a wire nut connector; and use inline conductor instead.