The type of game you hunt determines your choice of ammunition and choke more than your choice of gun, but each has their advantages and disadvantages.
Pumps and semi-autos are actually quite similar in their function and use, in that the feeding mechanisms and magazines are similar, and the trigger is pressed to fire the gun, then after the action cycles, must reset by allowing it to move forward slightly, before the subsequent shot can be fired. The use of a pump action requires a longer learning curve than the semi-auto, since its easier to produce shooter induced failures to cycle the gun. All you need for reliable cycling in a semi-auto is to be solidly behind the gun when it fires. The forend of the semi-auto it rigid so you might find it produces a more solid shooting platform that the "wobbly" forend of a pump. Given guns or equal quality, the pump will be less expensive than the semi-auto, both are less expensive than double guns or similar quality.
Doubles, that is O/Us or SXS guns, are viable options as well and should not be discounted if you can afford a good one. The prices range from the bargain basement level to the sky is the limit. A quality SXS double gun is as much a work of art as it is a practical hunting piece. There are those who prefer O/Us and those who prefer SXs, but something to consider is that the O/U must open to the limit of it's hinge to provide access to the lower barrel whereas the SXS need only open enough for the ejectors or extractors to function. You get to choose between ejectors and extractors in quality guns. You get to choose the grade of wood. You get to have the gun measured to you. Most of my shooting life has been directed towards rifles and handguns, but I do lust for a high quality SXS 20 ga.
Considering a good used gun is a way to step up in quality without paying an exorbitant price. This can be a stressful situation for the novice who doesn't want to be cheated, but doesn't think he knows enough to make a good choice. If that's you, have an experienced hand look the gun over; the seller should agree to a 3 day non-shooting inspection if the gun is bought on line. First look the gun over and make a determination of whether it has been cared for or abused. The outside of the gun is an indication of the condition of its insides. Beware of home gunsmithing "improvements". Beware of muzzle damage, barrel bulges, or other obvious damage. Be less concerned with handling marks and wear. Hinge action guns, singles or doubles, should lock up tight. The forends should not be inclined to pull free without effort, and the better ones have a recessed mechanical lock in the underside of the forend.
The choice between 12 ga and 20 will get different responses from different people, but my take is as follows. For the purposes you've outlined, I would look at a 20 ga. The guns are physically smaller and lighter than their 12 ga counterparts, so are less fatiguing to carry. While this will result in greater recoil than from a heavier gun, with the type of loads typically used for small game hunting, recoil is not a concern. If a 12 ga and a 20 ga load had the same shot weight and the guns had the same degree of choke, the 20 ga pattern would be tighter since the 20 ga barrel is smaller in diameter relative to the 12 ga.
