I would second the idea of going to the skeet or sporting clays range. From what you said you are partly trying to use rifle type thinking and techniques which will mess you up shooting a shotgun. They are entirely different activities. There really shouldn't be a "sight picture" as such with a shotgun. If you are seeing the barrel and bead when you shoot you're doing it wrong. Some shooting instructors will take the bead off when starting a new shooter. Your head is really your rear sight and gun aligment comes from shotgun fit, not sight picture. Your attention and focus should be entirely on your target and on your movement, not on the gun at all. With a rifle the sight picture at time of firing is all important. In a shotgun having the proper movement with a gun that shoots where you are looking is what's important. Have some one watch you shoot. If you are trying to "aim" the shotgun your swing will be a series of starts and stops as your attention goes back and forth from barrel to target. Rifle shooters often have some things to unlearn and can have trouble letting go and just swinging and shooting. It seems at first as if you are kind of "out of control" when you don't aim but precision shotgun shooting comes from something else. I know it seems expensive but you might even consider a lesson from a shooting instructor to get you started right so you don't develop bad shooting habits that will be hard to break later.

As far as hunting start with your state DNR website. There is a lot to learn but fortunately you don't have learn it all to start having a ball. Welcome.

http://www.tpwd.state.tx.us/