Hi Chita,
Welcome!
I'm another professional computer geek who also dabbles in A/V. I'll try to point you in the right direction here.
There are actually several main parts to the system you've described. One is a receiver with multi-zone capability, the other is a control system. In most cases, you'd also need some additional channels of amplification for the three extra zones.
There are several popular choices in control systems, with Control4 and Crestron being two that I've heard of frequently. These would most likely make up the "Brains" of the system to which you refer, including turning on or off the required electronic components, switching in the right sources, and so on.
Do you have an existing Home Theater receiver? If so, what brand and model? Where do the pre-wired speaker wires terminate? Are they all in the ceiling? Is the wiring panel that you mentioned in the great room? Is there some other wiring closet in which you can install the required equipment?
At a guess, the cost of all the required equipment to get this system working is likely significantly more than $2K even if you are able to install it yourself. As a compromise, you might consider just installing the entertainment system in the great room but this of course wouldn't give you any sound in any of the other rooms. However, it would greatly simplify the installation and could most likely be completed for under $2K depending on your choice of speakers, etc.
