In the O/U realm, the basics from Browning, Beretta, and even SKBs, (most likely used), will be your best bet.
Trap and skeet are two very different games and the guns are typically very different on certain aspects - trap guns are designed to shoot high so you float the rising target while skeet guns are designed to shoot flat so you cover the target and pull the trigger. A good O/U can work, but you will compromise a little in either or both games. If you are not using it for competition, then it won't matter much; but if you find yourself shooting competitive registered targets, that one here, one there, you missed means finishing WAY down the list.
Whatever you decide, do yourself a favor and get the target version and not the field model. Field guns are usually lighter (bad for recoil), shorter barrels, (bad for swing dynamics), but the best way to determine what will work nest for YOU is to go to your local gun clubs and rent/borrow/try as many guns as you can - even the ones way out of your budget. The reason being, if you find certain features on those high-ends guns (Kreighoff, Kolar, Perazzi, etc.) then you will have an idea of what to look for when you find guns in your budget. Things like chokes, gun weight, barrel weight, stock dimensions can be easily changed after the fact. Barrel length, perfect lock time and similar, not as easy
The fun is in the search - have fun doing it