Here is my trap gun:
It is also my skeet gun, my sporting clays gun, my hunting gun, and serves sometimes as my home defense gun. I paid two hundred and some odd dollars for it, brand new (of course that was about '98 or so). It's all you need to start. I've never had trouble cycling it for doubles, and have never gotten tired of pumping it on the trap line. It's certainly no worse than breaking open a break action every shot.
I shoot full choke for all trap, but I don't shoot a lot of 16 yd any more. I enjoy skeet more anyway. You don't really even need choke tubes. A lot of guys claim fixed choke barrels pattern better, although if you want to have one gun to do everything, tubes are nice.
BTW, anyone who says pumps are no good is a fool. Of the guys at my club, 3 of the best (the ones who always go home with more money than they brought), shoot a pair of Rem 870's and a model 12 Winchester. It's about getting the mechanics and lead down. Having a gun that fits well helps this, but any gun can be fit around you. A used Mossberg will do you just fine.
I would recommend a fairly long barrel: at least 26" and up. It simply makes moving the gun easier. It sounds counter-intuitive, but with a short barrel it is common to stop moving because it has less mass. A longer barrel is harder to stop, and gun movement is what shotgunning is all about. This matters a lot less in trap, than in skeet, however as there is a lot less movement. A longer barrel will also maximize velocity, and gives a longer sighting plane (although you aren't supposed to be looking at the sights, anyway, but that's another story....)