I agree with most of the posters that $2000 isn't enough to start a collection of "collectable" firearms. So why don't we consider "classic" firearms instead? That might fit within the budgetary constraints. He already has the shotguns, so....
Rifle - Pre '64 Win. in .270 or 30-06. This will cost ~$1200.
Pistol - Older S&W or Colt revolver in .357 Mag. This will cost ~$500
Rimfire Rifle - Marlin, Winchester, or Browning lever action. This will cost ~$300
Each of these will require a considerable amount of shopping to get a quality firearm at this price. By the time he's done, the OP will have both an education, and the start of a very nice collection.
None of the guns are so special that he can't take them out and shoot them or hunt with them. With normal care and use they will all outlast him, and keep their value.
Re-read the OP's comments further down in the thread, and since he doesn't want to shoot them, I would change my suggestions. I would favor handguns, since he wants to display them. Pick one that you find interesting and has been around for awhile. Lugers, 1911s, Browning Hi-Powers, S&W revolvers..... Any of these will have enough depth to keep you interested for the rest of your lifetime. And they will all look great in a display case in your game room.