Trap and Skeet are two different games, and have somewhat different criteria. Trap guns are adjusted to shoot high, since targets are rising, while skeet guns(like field shotguns) shoot dead on, at least in theory.Trap guns run a bit heavier,also.
I'm no expert at either, nor clays, just a generalized shooter. The most oft used clays gun tends to be a semi auto,with the Remington 1100 the most popular. O/Us tend to be Brownings, Berettas, etc, and are almost as widely used as the autos.
IMO, a good way to start out would be to get a semi,probably a used 1100,with choke tubes. Set the stock so it patterns half a pattern high,and go play. After burning up some ammo, I think you'll have a better idea of what works for YOU.
Also, instead of going for expensive and possibly useless mods and addons, get some professional training. A good shotist can shoot well with any clunker, the key is form, fit and practice.