Back in the day, way back, I asked the same question at one of the local firearms stores. The short answer was grab your shotgun and come out to the club on Wednesday after supper. So, I grabbed my shotgun, and some ammo and went to the local trap and skeet club. First off, Trap is not Skeet, two very separate games, about all that's similar is the ammunition, sort of. Trap is a game that imitates a pheasant rising up and flying away from you. 1 1/8 oz of 7 1/2 shot was a good load, some use #8. It is all 12 ga, and the shotgun is like your goose gun on steroids. A single shot or an O/U if you are shooting doubles.
Skeet is more like shooting chickens, going away, left to right or right to left, and at you. The birds are shot at shorter distance, the shotgun is usually an O/U as a few of the stages have doubles. It is a four gauge game, 12 ga, 20 ga, 28 ga and .410 cal. You don't need a 1 1/8 load. Here most use #9, but #8 and #7 1/2 are just as popular. You can get started with 1 shotgun, but in order to participate in a four gun shoot, you'll need at least 2, a .410 for .410 and 28 ga, and a 20 ga for 20 and 12. It was common to see shooters use a .410 for 28. Back in the day Browning and Winchester made 3 and 4 gun sets. The 3 gun was sweet, a set of .410 barrels, 28 ga barrels and 20 ga barrels on the small frame 20 ga shotgun.
I found out fairly quickly a large pail of money is handy to have, maybe several pails, as they tend to empty fairly quickly.
I've been out of the game for several decades, the information I gave you is old. I'm pretty sure the pail of money one still applies. the trap game is more expensive than the skeet game, but both will empty your pails. Trap just empties them faster.
Just for S&G I checked out the price of a new Perazzi shotgun, yep they are pricey, $20,000. If you are serious about trap, you'll need 2. There are other shotgun brands, but they're nothing like a Perazzi. If you want to shoot 100 straight fairly often, you'll need one.
Skeet is a bit easier on your wallet for shotguns, cut the chokes off any old O/U and go have some fun. Oh, the trap boys will snicker at you, but they're usually snobs, not people you'd want for friends anyways.
None of my reply was serious, although there is probably some truth to it.
I got out of the game while I still had 1 small pail of money left. Fun times.
Seriously, if you just want to have fun, and sharpen your shotgun skills, Sporting Clays is something to look into. Smaller pails, fewer snobs, and a bit more action.