The main clay games in the US are: American trap, American Skeet, 5-stand, and sporting clays. Also around is International Trap, aka Bunker, International Skeet, FITASC, and helice.
American singles trap is a unique game unto itself as only one round per target is used, (except for the time they shoot doubles, most folks don't), therefore the development of the trap single. Skeet has doubles, 5-stand, sporting, FITASC have doubles, International Skeet and Helice allow two shots at a single target.
Towards that end, to allow you the most flexibility, a gun capable of firing two shots would be best. While not necessarily the best for American trap, it can be done, and it doesn't sound like you are entering serious competition at this time.
Depending on your budget, I would recommend either a quality semi like the Beretta in one of the 390/391 flavors, or a good over under from the likes of Browning/Beretta/SKB.
The semis will be less expensive than the over unders. Even new semis will be less than a comparable used over/under. The O?U has the advantage of being able to use 2 different chokes for clay games or birds, while the single barrel semi means a compromise. It's not THAT big of a deal in most cases.
My suggestion would be to go to your local gun club (shotgun, not metallic), and explain your situation, ask to borrow or rent as many different guns as you can and shoot them to see which ones fit you the best and give you the score from the get go. While stock dimensions can always be tweaked, if there is something already really close, that would be the best place to start. A GOOD quality gun will not be cheap, but then you only buy it once (read my sig line). The money you will spend on ammo and targets, gas and hunting licenses will far outrun the cost of a good gun over your lifetime. Another advantage is that if you decide to stop, or upgrade to something better, you'll get most, if not all, of your money from a better quality gun than you will from the likes of the cheap ones.
there will be folks coming on here and saying how their Chinese/Brazilian/Russian gun has beaten the guys with Kreighoffs, etc. Most of them have never handled a well-balanced gun that runs forever - there IS a difference, and once you feel it and experience it for yourself, you'll understand what I am saying.
Good luck in your quest - if you give us some more info regarding budget, action type you like, recommendations can get narrowed even further
