The best first gun IMHO would be the .30-06. Others have recited most of the rationale for this recommendation, but I will restate them.

The cartridge can be used for all game animals -- and has been used successfully to take a pretty good portion of the big game animals that have been taken in North America over the last 50 years. Granted, other heavier cartridges may be preferred for brown and grizzly bear, but this does not change the fact that .30-06 has probably taken a lot of the brown and grizzly bears over the last 50 years. Magnum cartridges may be flatter shooting, but this difference probably will not make a dime's bit of difference until the shooting gets out beyond 300 yards, and most hunters, myself included, probably should not be taking shots out beyond 300 yards anyway because they don't shoot accurately enough under field conditions at 300 yards to reliably place their shots on game animals where they need to be placed.

The cartridge is omnipresent. I imagine you can buy a .30-06 rifle in any action -- lever, pump, auto-loader, single-shot, and bolt action. I imagine every manufacturer offers at least one rifle chambered in .30-06. A wide variety of .30-06 ammunition is available, from 125 grains up to 220 grains. If you can buy centerfire ammunition in a store, they will probably stock an .30-06 load appropriate for the game animals known in that area.

I have heard people say everyone should have a .30-06 in their battery of guns. There are other cartridges that are better than the .30-06 for specific purposes -- maybe the .25-06 for pronghorn antelope, for example-- but the .30-06 is a good back-up gun formostgame. And in most hunting situations it is a good idea to have a back-up gun if possible.