[legion]

As a guy who owns a single shot, an over/under, a pump, and a semi-auto, all in 12 gauge...

For trap/skeet, can't go wrong with an over/under. The main advantages are that you have a quick follow-up shot, have the same sight picture for both barrels, and the extra mass makes for more even, consistent swinging (and kind of forces a follow-throw). They are generally the most mechanically and most expensive kind of shotgun.

Side-by-sides are mostly relegated to cowboy shoots these days. They were originally designed to allow for a follow-up shot, but the the need for external hammers made an over/under impossible.

Pumps are simple and dependable. They are great for hunting. Most will tolerate some abuse.

Semi-autos have more capacity than over/unders, and can be much lighter. (Which can be both good and bad). Some (most) can be fussy about cleaning and ammunition. They can be pricey as well. If you get a gas-operated one, it will help reduce recoil as well.