The Shotgun (aka: “shotty”, "gauge" or “bitch”) is probably the most versatile firearm platform available but at the same time they are unforgiving. I have seen seasoned shotgunners look really awkward after just a couple of months out of practice.
You are dealing with a very violent firearm that is trying to feed, fire and expel massive shells with a flat top and squared-off edges. It’s almost like they were designed to jam during fast-fire. To truly be effective you need practice… lots of practice. Unless you are willing to put in some serious trigger time, forget about this option. If you are, then you can become very formative.
By the way, pistol grips without a stock are bullshit. Those are for suckers and posers. To be able to control a shotgun you need a shoulder stock… ones with integrated pistol grips are great too.
A good shotgun doesn’t have to be expensive, some high end models get a little ridiculous. There are two main categories, pump-action and semi-automatic (also called an auto, semi-auto or auto-loader). Double barreled models are tougher to master so I wouldn't bother. With a pump, you have to rack a round into the chamber before it will fire, an auto does this automatically after each round is fired.
Most people start out with a pump and there are many good ones on the market. If I had to choose one for a new shooter, it would be the Remington 870. Produced since 1949, they are dependable, reasonably priced and very popular. Popularity in the gun world means there are many aftermarket parts and upgrades available. These are great for hunting, skeet, trap or any other shotgun discipline. For home defense, I would stick with a shorter barrels, which is considered tactical lengths 18.5”- 22”, to make it more manageable in close quarters. Wood or plastic molded stocks are just a preference, either way it’s the same gun.
Run-and-gun events are best way to learn how to become effective with a shotgun. Shooting at paper targets while standing still is fine in the beginning but running around and engaging multiple targets is the best way to become truly bad-ass.
Watch some Tactical Shotgun or 3 Gun Nation events on Youtube and you will see how awesome people can become.