You can get a single gun that will be damn good for skeet and sporting clays, but it will be a good bit less than optimum for trap. Trap requires the most specialized guns of the shotgun sports. All trap birds are taken on the rise. Your lead will place the muzzle of a neutrally set up gun over the birds, blocking them out. Most shooters don't like that. So trap guns have high combs and ribs that are high in the back so that they can lead the birds and still see them. They also like really long barrels, even as much as 34" on an O/U. This setup isn't good for the other games.
In the end, most sporting shooters go with a O/U. They like the balance, and like having two chokes. But a noticeable minority use a gas auto, almost all choosing a Beretta 391. In an O/U, they like 30" and 32" barrels. In an auto, some will use 28", but many like the 30" even with the longer receiver of an auto. Either gun will be a good 12 gauge skeet gun.
But if you get into "real" skeet, you have to shoot all the gauges. Some shooters shoot the 20 in the 12 gauge, thinking the lower recoil more than offsets the fewer pellets. I am not a serious skeet shooter, but I agree. I hit the same number of birds with either, and the 20 is a lot softer. Skeet shooters like O/Us with replaceable tube sets to shoot the different guages because this gives them the same gun feel in every gauge.
In short, you can do quite well for 12 gauge skeet and sporting with one gun. (Sporting shooters all use the 12 - there will always be a few shots where you want every pellet you can get.) That gun can be used for trap, but will be more at a disadvantage there, if you get serious.
Look for a used gun. You can get a good used O/U for $1000 or $1200. They last forever and clay shooters trade guns frequently. You can probably find a Beretta 391 used for $600 or $700 if you look.