I recommend a combo shotgun that comes with a smoothbore barrel that accepts different choke tubes, and a rifled slug barrel with a cantilevered scope mount.
With the various chokes you can hunt all small game, turkeys, ducks and geese, etc. The slug barrel is obviously for deer, but other big game too like black bears, boar, heck even moose.
If you decide to stick with gun hunting you can buy other guns for more specific purposes, but the combo shotgun is the cheapest way to get into it, and is still a gun you will hang onto for its versatility.
Mine is a Mossberg 500 that came with a smoothbore slug barrel/iron sights. It is good out to 75 yds for deer. These days most combos come with a rifled barrel. The remington 870 is another good choice. Pick one of each up and see which you prefer.