RE: new slug gun
I just did a ton of research myself because I was in the process of buying one myself. I have a Browning BPS fully rifled barrel, and it is not a bad shooter, but not great. I wanted to move into a bolt action to improve my groups, especially out to 100 yards. Some of what I am going to tell you comes from first hand experience, but most of it came as a result of a seminar I attended by Dave Henderson. Dave is described in the industry as a "specialist" in slug hunting having started as a kid and has shot professionally and competitively for the remainder of his life. So, my first suggestion is to buy his book which is available on Amazon.com and is titled Shotgunning For Deer by Dave Henderson $24.95. He will explain in avery easy to read and understand format allyou need to know.
If I may paraphrase some of his points for youit would be as follows. Understand that the barrel makes a difference. While 60% of shooters today are still using smooth bore barrels, rifled barrels are becoming more popular, particularly with new slugs being developed. A smooth bore in general (and I don't mean to imply always) is going to be effective to about 60 yards. A rifled barrel may get you to 100 yards...possibly 125 yards, but at that range it takes an expert shooter and ideal conditions to be proficient. Understand that your slug will not travel like a rifle bullet because it is so much larger and heavier. In a gentle breeze of 10 mph, your slug will drift 6-8 inches at 100 yards. So you can see that getting good groups at 100 yards is difficult outdoors, which is why it is suggested to sight them in at 50 yards an 1 1/2"-2" high. Obviously choosing the correct slugs will be important too and this is Dependant upon the gun you choose. Smooth vs rifled and the rate of twist are important to understand and know what slugs work best for that setup.
Having said all that to get back to your question of what is the best type of gun, Henderson says (and I have heard this to be true most of my life also) that in general (some people will have an exception to this as always) auto loaders are the least accurate slug shooter and this is because of the gun's action. With the bolt moving at discharge, the motion affects the barrel slightly before the slug leaves the gun. Pump guns would be next best in general, but the bolt does not lock like say a bolt action and also the barrel may be subject to minute movement and vibration from the discharge at the point where the barrel slides into the receiver. Henderson recommends having the barrel pinned to the receiver whenever possible. Third, and again, in general, the bolt action has always been considered to be the most accurate, I believe mostly because the bolt is locked in a fixed position and cannot move. Everything is in a straight line and so they are generally considered to be the most accurate slug guns.
During my search for a new slug gun, I was only interested in a bolt action 12 Gage. But please take note that the 20 Gage is becoming very popular and ballistically, it has many advantages over a 12 Gage including a much reduced recoil so you might want to consider a 20 Gage. There are 3-4 slug, bolt actions to consider. At the top of the heap of guns in production is the Tar Hunt. It is the Cadillac and has a price tag to match. A new one will set you back over $2500. I did see a used one go on gunbroker.com for just over $1800 recently though. The Browning A-Bolt shotgun is considered to have been the best mass produced bolt ever built. It was only built for 3 years and because of the high price (over $700 new)and low sales, Browning discontinued production. If you can find a used one you will pay a hefty price, but have a great gun. I have seen new fully rifled versions (don't get a rifled choke tube model) go for over$2000 on the gun auction sites with used ones in good condition go for $1800. The next model is the Savage 210F. Henderson says this is one of the best shooting (in terms of groups) and it is only a fraction of the cost of the others. They are still in production but I believe Savage does not make a whole lot of them each year so you may have to look for them. I just bought a camo version of the Savage 210 myself. It was the last one the dealer had and I paid $470 for it. There is also a Marlin, but I don't think it is still in production. The Marlin 512 I have heard is a good accurate shooter and I see them frequently on the gun auctions for $350-$425.
A final note I would make would be about recoil and the trigger on a slug gun. First the recoil can be almost horrendous at timesif you're shooting a couple of boxes at the range. Last year after 12 shots out of my Browning BPS, I had a very badly bruised shoulder for 2 weeks. Adding weight and a good recoil pad can help you out there. Secondly, the triggers on these slug shotguns are the same ones used for the bird shot. There is a ton of creep and the pull is sometimes close to 8-10 pounds. Consider having a trigger job done to make the trigger crisp and a pull of maybe 5 pounds to preferably 3.5 pounds. Good luck with your search.