Which game suits you best depends on your personality, ability and budget. Those who get into shooting in their retirement years may not have the eyesight nor reflexes required for the faster games, so trap is their favorite.
What's available locally is also a governing factor -- you can shoot only what's offered. For the club, it's a matter of economics: A trap field is the cheapest to build and operate. And, trap takes the least acreage, followed by Skeet and sporting clays. When sporting clays was starting, we wanted to set it up at our club, but we didn't have the space. So, we compromised and installed 5-stand clays on two of the overlaid fields.
Skeet is my favorite, but I've competed in both NSSA and ATA events. Among Skeet shooters, there's a saying: If you learned how to get along well with others in kindergarten, then Skeet is the game for you. If you didn't, then you might be better suited to trap because they keep the shooters separated on their own little squares, and you don't have to share. You can draw your own conclusions why sporting clays are shot from cages.
Many folks start shooting the clay sports with field or general purpose guns -- some entry level pumps. If you find yourself attracted to the games, you'll soon want to get a target gun. My first gun was a budget pump that was soon abandoned in favor of an Japanese O/U. Over the years, that first O/U was superseded by several custom O/Us.
My friend TheKlawMan is another example: He came to TFL wanting advice on getting a shotgun. He purchased a R-870-Ex and used it to get acclimated to shotgunning. The acclimating soon became an addiction and he started reloading his own ammo. These days, he's a range regular with his new O/U.