I can't add much to the long range shooting part of the site as a real nubie, but I have been shooting competitive shotguns since the 60s.
I have O/Us, SxSs, Pumps and Autoloaders. For trap and skeet (as well as sporting clays and 5 stand) you will find that O/Us rule the day. Not that autos don't shoot any worse, they shoot just as good as my K-80s, but throwing empties is one of the pet peeves of many trap shooters and for me the cycling of the action and throwing of the empty is a distraction in doubles. Shooting clays normally means a lot more rounds fired than for a hunting gun. Pumps, though they were popular for skeet in the early days (Model 12 Winchesters especially), pumping is too much to remember these days so not so good for doubles, though if you are only going to shoot singles in trap, you will do just as well with a pump Remington 870 trap gun for a lot less money.
In the under $2K range, I would lean toward used guns that are proven to make it through 10s of thousands of rounds. As mentioned, the Beretta 682 and Browning Citories are amoung those. Keep in mind that the desired Point of Impact for trap gun is different than that for a skeet gun, so if you are using one only, get one with an adjustable comb. An adjustable rib would also be a benefit, but most of the O/Us with adjustable ribs will be over your budget even used.
You could buy two autoloaders for under $2K - and have them set up for eash POI. Remington 1100 Sportings are popular and than you don't have to mess with changing locations of adjustable combs. They are available in left hand versions.
A neutral cast will allow you to shoot left handed (make up for it by offset of the adjustable comb) and sell it later to a right handed or left handed shooter.
The newer 682 and Citori's have screw in chokes. 682 Golden E, Citori XT, Speacial Sporting etc. You might find a good deal on a slightly used Browning Crossover.
Jaqua's in Finley Ohio and Southwest Shooter's Supply in Phonex, both have a lot of used guns and though you will pay a little more than if you buy from the users, these stores, as well as others, will give you solid information and condition reports and are reputable to deal with. Give Bruce Kinkner a call at Southwest and he will talk to you for hours about shot guns.
All this being said, a lot depends on if you want to shoot serious clays as FIT is the number one area required for success. A good shotgun smith can help set up a gun for you, but nothing beats borrowing some guns to try first, especially on a patterning board. Most guys I have met at the local clays ranges are happy to help. Most will be happy to have you shoot their guns. Saves a lot of money in the long run when you buy something that turns out not to be the right one for you.
From a gauge standpoint, my best averages are in 20 gauge, followed by 28 and then 12. I never thought that I was recoil adverse, but apparently I am more than I know. Depending on how heavy a gun you can tolerate, a 12 qauge with light loads will work as pointed out and you can always add 11 ounces and a set of ultra light tubes in 20 or 28 or .410 if and when you want to. If you want a lighter gun, then a 20 gauge on a 20 gauge frame might be the best bet as you can also shoot 1 ounce 20 gauge in 12 gauge events.
Bob