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As a life long hunter, I think you should run what you have. Shooting skeet from a low gun with the safety on is great practice for bird hunting. It'll cover incoming, outgoing and crossing shots that are relevant to waterfowl or upland gunning over a flushing dog. Trap relates more to hunting over a pointing dog.If you need a new gun, look close at Browning or Beretta for over/unders. If you go with a cheap over/under, you'll start looking at Browning/Beretta guns in a few years.
Browning, Berretta, Winchester 101, Weatherby Orion, SKB, Ruger Red LabelAny one of these will work just fine. I would not go with any thing cheaper. The secret to shotguns is the FIT. If the gun do not fit you, you will miss a large % of your shots.