First up understand there is no "BEST" brand or model for all round or any other use. They are all compromises and what might be "best" for you might not be for me or the next guy. Having said that let's address your uses and what might work for them.
Trap is a very specialized game. All targets are shot going away from you and the gun is generally premounted. All those targets are rising also. So the guns made for use on trap fields have a few things in common. They almost always have longer than standard lenth of pull that might present a problem when hunting in cold weather as mounting them might not be fast or smooth. They have a higher POI in relationship to POA to compensate for the rising targets. Now this isn't necessarily so bad if what you're shooting is flushing birds. But on crossing shots it can be. Trap guns in general don't make good all round guns.
Skeet is a totally different game. Targets are all shot far closer to the gun and are flying reasonably flat when shot. You want a gun that has a POI very close to the POA. Generally speaking you'll not shoot as well on a skeet course with a trap gun as one made for field or skeet or even for sporting clays.
Sporting clays is a game that has a wide variety of target presentations and each course you shoot will be different from all others as well as each station being different from the others on the course. You'll see incomers, outgoers, straight up targets and targets falling like a rock, crossing targets and all sorts of ranges and presentations. Guns for it are usually slightly modified field/skeet type guns but generally with very long barrels. Mine has 32" barrels. Few who have dedicated guns for this game go shorter than 30" with their barrels.
So if you're going to be serious at either of these games you'll want a gun specifically intended for it. If you just want to play at them for fun and to improve your field shooting them get a field or skeet gun and make do. Unless trap will be shot seriously for competition then I'd not get a dedicated trap gun at all.
For an all round gun to be used for all uses, field, trap, skeet and sporting clays I'd get a gun labeled by the maker as a skeet or sporting clays gun. Get 28" or 30" barrels and a variety of choke tubes. You'll want a pair of tubes marked skeet, an IC, a MOD and maybe a Light Mod and a Full at least. Use the gun as is for all but trap and for that you might buy a slip on or tie on leather cheek piece to raise the comb to suit it better to the game of trap. But if you're only playing at it and don't care if you reach your absolute best at it then just use the gun as is and don't worry about it.
The Browning Citori is an excellent gun and perhaps the most popular and best selling of them. Beretta is also fine and generally a bit lighter. Today there are so many names out there it's hard to settle on just one. But really Browning and Berreta seem the most used and especially so in the price range you have mentioned. Ruger will work OK also.
Most folks who are serious shotgunners will have a variety of O/Us for the various uses to which they will put it.
