12 gauge pump
I would suggest a 12 gauge pump. Depending on your budget and likes there are many models to choose from. Also, these can be picked up used for good prices. One of the reasons I like the pump is it "always works" regardless of the ammo.
Some semi-autos don't function well with certain ammunition but the pump functions because you, the operator, are the one making it function. You can shoot light or heavy loads, it simplly doesn't matter.
Getting into semi-autos can get considerably higher in price and is about like anything else in that you get what you pay for. You can easily spend $1,500.00 - $2,000.00 on a good semi auto.
Some will tell you to shoot 3-1/2 inch shells and I've done it. BUT, in my opinion, it is brutal.
The recoil in a semi-auto will be a bit less than the pump but in a real world hunting situation, I'm not sure if that makes a considerable difference. This year I'll be hunting with a gel pad on my shoulder.
The gun will cost more if it has some sort of manufacturer's "protective coating" or is done in Max HD camo, etc.
Good luck in your search. I'm sure you'll get plenty of great advice from the many experienced shotgunners here on TFL.