If you're looking for just 1 gun to do it all then keep in mind that regardless of what you choose it will be great in certain situations and lack in others. My opinion is to get a smooth bore shotgun. They can shoot all loads that you will need for all 3 types of hunts.
Deer - you'll need a slugger type ammo (pre rifled). Accuracy is not as good but it will do the job
Bird - with bird loads the groups wont be as tight but once again it will do the trick.
My first shotgun was a 12ga Mossberg 500 Ducks Unlimited and it did well for deer, pheasant and quail hunting