Prius starts moving when first placed into gear with foot on the brake

Overall this is serious. I would take it in for proper diagnosis.

I will assume you have no warning lights up that might point you in the right direction. You could power cycle everything by disconnecting the 12v battery for a few minutes and then test. But the first thing I would do before cycling power is to use a scanner and check codes. Power cycling will clear them making further diagnosis harder.

Then I might check the brake switch status to ensure the system is seeing it properly. Remember this is a brake by wire car so the ecus have to know when you are trying to brake. There is also a brake pedal stroke sensor.

Even though brake actuation is computer controlled, it still operates a hydraulic system fed by an electric pump and actuator. You would normally have warning lights and codes if the actuator was not functional. If the hydraulics had faults such as air in the system or an internal master cylinder leak, it might be possible that pumping the brakes temporarily moves the brake cylinders.

There are two calibrations that can be done of the brake system with the brake pedal stroke sensor sometimes causing trouble. The brake actuator/pump is also a frequent failure.

Again this is serious. I would take it in for proper diagnosis.