I handload and shoot a lot of different calibers, and I would also recommend getting a 12GA instead of 20GA. You can buy plenty of 12GA ammo that is loaded to nearly identical specs as 20GA. (The 7/8oz and 1oz trap & skeet loads fit into this category). With 12GA, however, you also have the option of much heavier loads for geese and turkeys, as well as various types of slugs, larger buckshot, and even things like signal flares (must be fired from an open choke!) or beanbag rounds used for chasing off bears.
If you stick to one of the standard, reasonably-afforadable guns (Remington 870 or 1100, Mossberg 500), you can also swap out the barrels for different types of hunting/shooting. Most trap and skeet ranges won't let you shoot less than a 24" barrel, and slugs work best from a dedicated slug barrel if you're going to go deer hunting. You can also get shorter barrels for defense or use in heavy brush. The barrels are cheap, and you can often find various used barrels for sale in the Winter, after hunting season is over in most places.
You won't be able to swap barrels or use the full variety of shotgun loads if you go with a 20GA in a fixed config.
For the rest, I would also say you should just get a 22 rifle from any quality manufacturer. A lever gun with a peep sight is a lot of fun and has a neat, historical look. A bolt or semi-auto will also work just fine as well.
Arclight