Don't load that shotgun until you have an experienced mentor standing beside you, outside on a range!!!
Take all advice you get on the internet with extreme skepticism, including this post.
Learning to shoot all by yourself is not the best way. Same goes for skydiving. You want lessons.
BE SAFE.
Most of us were taught by family as kids, then took excellent State Hunter Safety Classes so we could get a hunting license in middle school or high school. Many people learned in the military. Then we all get instruction from range safety officers. SAFETY. Number ONE is safety. How to hold it safely, move with it safely, load and unload safely, drill in to us how to have ABSOLUTE control over where the barrel of that gun is pointed ONE HUNDRED PERCENT of the time. It's NOT EASY, even for us old dogs.
THIS IS DEADLY SERIOUS BUSINESS
In your quest for "home defense" you just brought a weapon you have no skill or knowlege of in to your home. There are so many ways this plan can and does go wrong, with deadly consequences.
We have been doing this so long, most of us forget what it is to start out from scratch. That's why a good instructor is much better than a friend.
Learn to follow the rules of gun safety instinctively as putting on your pants in the morning.
1. Look up "gun safety" on Wikipedia. Read it, learn it, copy the rules in to a notebook ten times. It's not complete but it's a good start.
2. Read the manual for your Mossberg 590 riot gun (6 shell magazine cylinder bore (choke) 18.5" barrel.
3. Look for a local shooting club or large private range to get lessons at. Talk to them before buying ammunition, they may have specific requirements. Or find an OLD friend that really knows what he's doing and can take you out and help you select what you need before hitting the outdoors to blow some stuff up.
4. Order yourself 3 or more 20 gauge "Snap Caps"- these are special FAKE SHELLS you can use to practice with. They are special color and design. NEVER load live ammunition unless you intend to obliterate something. Seriously.
5. Don't be buying a bunch of "tactical" toys like peep sights, slings, flashlights that bolt on, pistol grips, collapsing stocks, bayonettes, waffle makers or any of that "tactical" crap until you know what you are doing. Then you'll have saved a lot of money for a competition or sporting shotgun.
6. Don't worry about little details like "Home Defense Loads". Worrying over what exact type of shell to use is like buying a car and asking "What brand of Oil should I put in it?" So long as it fits, it will obliterate anything at the range you can find inside your house.
*quick answer:
20 Gauge 2 3/4" Lead Shot TARGET (#8 or #9 shot) will work fine and you can shoot it at all shotgun clubs because the small size shot will only go about 100 yards. (If your house is bigger than that, just hire a bodyguard.)
20 Gauge 2 3/4" Lead Shot GAME (any size shot) is fine for shooting outside on PRIVATE land with permission of the owner. Consider that walking around with a loaded shotgun during hunting season on public lands will get you a stiff fine unless you have a hunting license. To a game warden, you look like a poacher.
Steel, Bismuth and Tungsten shells are for hunting because of complex ecological issues people debate about.
Buckshot is round balls the size of peas. Slugs are basically enormous crude bullets. Having either of those loaded while you are on public lands will get you a poaching ticket unless it's deer season and you have a deer tag- and that's SERIOUSLY BAD. Like take away your gun, car and freedom bad. At any distance you can come up with inside your house, the difference this stuff makes to an intruder isn't worth talking about. The lowly No.9 shot target load will make hamburger out of anything 1t 20 foot range.
3" shells are longer so they can put more in them, mostly for pheasant, duck or turkey hunting which is "long range" shotgun stuff. (40 to 50 yards.) They kick hard, you don't need that.
You have Cylinder Bore (or choke). That means your shotgun is a straight pipe. It's designed for shooting at about 20 yards range. This is marginally okay for the game of Skeet, which is pretty darned fun. People will look at you funny when you show up with that gun at the skeet line. Be absolutely sure to have an instructor with you before you even attempt it- for safety and to get invaluable tips on how to shoot.
Choke: to extend the range for hunting or games like Trap, shotguns are smaller inside the pipe at the front end than at the back end. Think of it as funneling down the shot in to a smaller stream size, even though scientifically that's absolutely not how it works.
Most modern sporting guns have a threaded front end and different size choke tubes are screwed in to vary how much "squeeze" the lead shot gets as it flies out, depending on what is being hunted or the shotgun game being played. Older shotguns have the "squeeze" at the front permanently drawn in to the metal and the shooter just lives with it not being perfect for every possible use.
You have a riot gun designed for combat and police use. You just have to live with "Open" choke.
1. Study gun safety like your life and the life of everyone around you depends on it. It does.
2. Read the manual for your gun!
3. Get instruction before even buying ammunition!
4. Get some 20 gauge snap caps when you pick up ammunition
5. Have fun. You have lived your whole life without a firearm and you're fine. Join a club, take some classes, have fun, don't buy in to this whole "self defense" thing. All of us sport shooters can operate shotguns safely and very effectively by instinct in pitch darkness. Most of the mall ninjas collect combat hardware and can't hit a thing or even get the safety off in times of stress.
6. Shotgun games and hunting are lifelong pastimes that are enjoyable on so many levels, look around for a shotgun club and be prepared to meet a bunch of really great fellas and some women that are pretty cool too.
Welcome aboard! Be safe. Have fun.