A new person hooked on QUACK!
The kind of gun you get depends on several factors.
How will you hunt? Open water, timber, fields, etc.
Steel shot is the most common shot. Other non-toxic fodder is available (at typically high cost) but steel works. How does this relate to the shotgun. You will need a fairly modern gun, past 15~ 20 years or a very high quality older gun for the barrel to be able to resist the hardness of the steel pellets. Shot wads have come a long way, byt steel does not "flow" like lead. It tends to cram through the choke in a wad of shot. My dad's Browning auto 5 circa 1969 has digested thousands of rounds of steel with no ill effect.
The secret to steel ids larger pellet size and SPEED. The lighter steel looses energy quickly. Pick up a box of steel and usually there is a shot size recomendation on the back of the box. I like 2's for ducks, I used to use 4's in lead shot. Keep the speed up to around 1400 ~ 1500 FPS. Standard loads are 1350. They don't cut it AS WELL. To get speed, and a decent payload minimum 1 1/8 oz. You would be served well with a gun that is 3" capable 3.5" is handy on geese. Practice with the gun you duck hunt with.
You need to choose from a pump or semi-auto. Pumps are less expensive. Auto's are good for quick follow up shots. I don't care how good you are with a pump. Get turned sideways, twisted up shooting at a mallard doing 75 over your head and the grace sometimes goes out the window. BTW never take an unsafe shot. Always know your surroundings with your hunting buddies and neighbors.
Pick up a lot of guns and feel which one is right for you. Swing, and balance are important. I like 28"+ barrels on duck/goose guns. Your probably not going to be carrying it in the field much. A longer sight plane is nice. I also find some shorter bbl lighter guns swing too fast for me. Personal preference all the way though.
Patten your gun. It dosen't have to be real scientific. You'd be suprised at the number of people who do not do this. draw a 30" circle on paper, pace off 30 ~40 yards and make sure your point of aim is centered and pellet dispertion is fairly even. Steel shot shoots tighter in most chokes compared to lead. try different chokes. Again, go to the trap range. Shoot, shoot, shoot.
Synthetic or wood? I like the look and feel of wood. Some guys claim synthetic being lighter transmits more felt recoil too. Cammo or black syn. It dosen't really matter as long as it's not shiny. Ducks pick up movement and shiny things.
Sling. depending on how your hunting you may want the capability to install a sling for the gun.
Price. Get the gun you want. if it's 200 or 900 dollars. The other acceaories you will buy will make the gun price look like childs play. unless you have a buddy who will let you tag along with his dog, boat, decoys, borrowed waders, clothes, heater. etc. etc..
If you have the opportunity, shoot some of your friends guns. Whatever you get, practice and make safety 2nd nature. We duck hunters do things that would make Mr. safety cringe, but it's part of the territory. Loading guns in blinds with more than one person. Shooting at cripple ducks on the water, shooting from a boat, establishing shooting zones in blinds so no one gets thier head blown off, all that fun stuff. Suprisingly the sport is pretty accident free. Mostly drownigs and hypothermia. Oh, find out where other duck hunters buy thier guns. The help will more than likely be more custom tailored there.
Foregot to mention doubles and over/unders. I don't normally consider them because tha are very hard to load in a pit, somewhat cumbersome in a blind. 2 shots are normally all you get though.
Now, let's get a shotgun and go hunting.