Like T The Old Jaybird said, a Remington 1100 with two barrels would work well for both skeet and trap. Barrel lengths would need to be about 28" and 32" respectively give or take a couple of inches either way.
If you get really serious about either sport you may want to consider an over/under shotgun, that way you can save your hulls for reloading. The semi automatic 1100 ejects them out of reach and many club shoots won't let you pick up your empty hulls. Believe me, serious skeet shooting means lots of practice, and reloading your own shells will make the sport more affordable for you if you plan on bringing home some trophies.
Whatever you choose, skeet or trap, enjoy it. You'll meet lots of friendly shooters and in my experience they always give a warm welcome to a "newbie" to the sport.
