Almost any shotgun can be used for shooting clay targets, but there are some features that make it easier. Unfortunately, the ones that make it easier for trap may make it harder for skeet.
A 12 gauge is the most versatile, so you are making a good choice there.
If you are looking for one gun to do it all, auto45 has the right idea as a sporting clays model is a compromise of several disciplines.
A longer barrel, 32 inches, is good for trap, a 28 inch is best for skeet, so a 30 inch is practical for both.
In your price range, I would look for a used semi-auto or overunder. The pump gun will work fine for trap, but you will need to learn to cycle it very quickly for doubles trap or skeet. You may find it necessary to incrrease you budget into the $500 range. While you can find good guns at $350, you will have many more options at $500.
The main features to look for are the shape of the stock, chokes, the rib, and how well the gun "fits" you.