Once again, let’s go back to your answer to why you want to get involved in shooting. The “right” shotgun all depends on your situation. For home defense, personally I am a fan of a pump-action shotgun with 00 (“double-aught”) buckshot. I like pump-action for its reliability, high-capacity, and simplicity, and 00 buckshot is a generally thought to be a good home-defense round, as it is less likely to punch through walls and create collateral damage. But perhaps you might consider a semiautomatic shotgun; it all depends on a number of factors (which are for another book).
For trap, skeet, and sporting clays, a break-action side-by-side or over-under are generally your main options.
With regards to caliber, I suggest 12 or 20 gauge for the beginner. Even though .410 bore is smaller and has less recoil, it is considered a caliber for experts, as the pattern is smaller and therefore makes it harder to hit clays. However, if you do not plan on shooting at clays and just want to shoot at paper targets, then .410 will work just fine.
With respect to home defense, 12 or 20 gauge is typically more appropriate, as either have more stopping power compared with .410 bore. However, at the end of the day, you should shoot the most powerful gauge where you will get accurate hits. There is no sense shooting a 12-gauge shotgun if you can’t hit your target.
Shotguns can cost anywhere from a few hundred dollars all the way into the thousands of dollars. As a general principle, I always say buy the best you can afford because you get what you will pay for. Another general principle is to decide how often you plan to use the item, and pay accordingly. If you’re interested in just trying a shotgun, perhaps just go rent a few or go to the range with some friends to try their shotguns.
But if you want to join the hundreds of thousands of shotgun owners around the world, I encourage you to join us.