Trap and skeet are very different sports, and what makes a good trap gun is different from what makes a good skeet gun. If you are planning to become GOOD at both, you are eventually going to want a different gun for each.
That being said, an O/U is a great choice for either. Make sure it has interchangeable chokes. For skeet, you want very little choke, as most of your shots are pretty close. For trap, you want a lot of choke, since ranges never get closer than 16 yards, and get as far as 33 yards.
You'll never take more than 2 shots at a time in either sport, so two barrels is just right.
In skeet, you swing the gun from side to side a lot, so a shorter barrel reduces weight and inertia, making those shots easier. Also, at close ranges the birds break easier, so you could opt for 20 gauge to save wear and tear on your shoulder.
In trap, it's all long range, so you need a tight choke with a good pattern. Longer barrels are generally preferred, but a short barrel with a good choke will get you out to 21 yards no problem. 12 gauge is essential to reach longer ranges.
So, look for an O/U with a relatively short barrel and interchangeable chokes.
Sports Authority used to carry a line called, I think,
Stoeger. They came in both 12 and 20 gauge. I looked at buying one many times because they were affordable, but I picked up a nice Beretta in a pawn shop some years ago and don't really need a second O/U.