Ok, a long question, so forgive a long answer.
First, a good gun. A Mossy ain't pretty, but it wil do the job and the tubes are reasonably priced. (Plus, the safety is on top where it belongs).
As to hunting:
Here's the general rule of thumb:
Dove and quail, you want improved cylinder and either #8s or #71/2s. I'd go with #8s as they are dirt cheap if you buy by the case at a big box like Gander or Bass Pro, and yes you will burn through a case speedy quick during dove season or shooting some skeet to practice, so buy a couple.
As far as small game, like rabbits, I use a modified cylinder and high brass #6. If you have to use steel due to local restrictions use #4s.
Ducks, where steel is required, go with high brass 3" #2. Its over kill, but steel is light and ducks take some putting down. Lighter shot means lost birds, particularly if you hunt in marshes. If you are oozing money and use Bismuth shot, #4s will do just fine. Again, modified is the way to go.
Turkeys and geese. I use BB, although folks swear by T shot for turkeys. I haven't shot one in years, as turkey hunting is up there with ice fishing on my list of things I want to do in the woods. I mean getting up at O dark thirty to sit in the woods swatting bugs to bushwack a bird I can buy a Publix for $12 isn't my thing. You're on your own here.
As for deer and hogs? I've never used buck shot. I prefer a slug as I just don't trust the shot for a clean kill. For slugs, you want either cylinder bore or a rifled tube. I have limited experience here as I use a rifle or bow if I can. I have killed a couple of hogs with a NE Arms single shot in cylinder using slugs, but others living in shotgun states will give better answers.
Good luck
FQ13