New to Prius. What's that noise?

Probably the ICE coolant being injected back into the engine from the thermos bottle. It does this to decrease the time before the engine reaches operating temperature. A similar noise is heard when you shut off the vehicle as the bottle is loaded.

 

The first noise is the sound of the brake pressure pump, pumping you up some brake pressure you can have on tap when stopping the car. It's doing what it's there for, which should make you smile.

Edit: or as AzWxGuy suggested, the pump moving coolant out of the thermos, though if it's that the recording makes it sound noisier than I'm used to. I was hearing the kind of joy-buzzery sound I associate more with the brake pressure pump. But the timing—seemingly after going READY—does sound more like the thermos pump, as the brake pump usually tries to get a head start on things by running as soon as you open the driver's door.

The second noise is also normal, if you mean that sort of whiny B-above-middle-C-ish noise. You can read about it in this post.

 

I retract my guess. My 2008 was a long time ago. There is a noise after shutting off the engine though, as the ICE coolant thermos bottle is loaded.

 

No, the thermos pump is a pump, and the valve is a valve.

They are related; the valve changes positions to change the coolant flow path between normal engine operation, saving hot coolant in the thermos, or getting it out. The pumping of the coolant is done by the pump.