If you go to www.trapshooters.com and ask this question you can get 100 answers in one day. However most of them will say that if you already have a decent pump you are good to go for all those sports.
A Rem 870 or if you must spend money a newer 1100 are great guns. Oh guess what? they just happen to make these neat-o shell catchers that hook onto the gun and catch your spent hulls so they don't fly all over the place and hit the guy next to you.. These are available for 870's 1100's and some of the Beretta Autos as well. More people get into clays shooting with Rem 870/1100's than all other guns combined.
There is no need to go ape here and by something you don't need . You need to see if you even like these games. I started Trapshooting a few years ago with my Browning A5. Then I bought a Citori XT Trap gun. Then I decided I liked Skeet better. Luckily that gun works for Skeet pretty good as well. However I really wanted a .410 for Skeet so I bought a Citori Sporting Clays Edition with 32" bbls for that and I don't shoot my 12 ga. as much now.
These guns were both $2000 ea. They along with Beretta Silver Pigeon shotguns are considered to be "Entry Level Clay Target Shotguns." They are real quality guns and will last thru tens of thousands of rounds and still be good to go. Citori's generally need the springs changed out around 75-100,000 rounds. As long as you keep them nice and don't scratch them all up or abuse them they hold their value well. It is not unusual for a quality Trap Shotgun to have 250,000 rounds thru it.
Or you can spend as much as you can put together,,, Literally. A $20,000 shotgun is a nice piece of kit, but it is not even close to being the top of the heap. I have a friend that just spent $53K on a custom built 28ga. Perrazzi that took nearly two years to build. He loves it. and since he is a dedicated shotgunner who shoots 3-5 times a week, he will get his monies worth out of the gun and probably sell it for what he paid or more. It is not uncommon to go to a large shoot like the Grand American and see guns that are worth $100K! +
So you see. If I was you I'd shoot those pump guns for a year and see if I like the games, and if you do then maybe look at a Citori or Beretta. That way you'll have a better idea of what it is that you really want as opposed to what we say you should buy. You will need to spend around $1800-2000 for the gun and then another $6-800 to get setup to reload the ammo so you can afford to shoot the thing enough, to get good enough, so you might be able to win some money. Or you can shoot your 870 for the rest of your life and have fun.
Beware of sharks in the water, because clay target shooting makes 9 Ball seem like a child's game. 2 years ago a 13 year old kid won the Grand American and $100,000! He didn't even make the cut last year. Lots of money in these sports.
Randy