Both the 870 and 500 are outstanding pump guns, you honestly can't go wrong with either of them! Both are very affordable, reliable, durable, will have more than enough capacity for your needs (as long as you choose the right model) and have plenty of accessories available.
Having said that, and fired both guns over the years, I personally prefer the Mossbergs, mostly because of the safety location. I'm a lefty, and the cross bolt safeties on 870s are less ergonomic for me to use. I also like how you don't have to push the shells past a loading gate to get them into the tube on the Mossberg guns; just drop them in and push. I also prefer the location of the slide release on the 500 as well.
Now in full disclosure I have a Remington 1100 (which I love!) and it has a cross-bolt safety and a loading gate I have to push the shells past and I can run that gun no problem. But when talking strictly Remington 870 vs. Mossberg 500 I prefer Mossberg.
The first shotgun I ever hunted was a Mossberg 500 20 GA. I took many squirrels with it as a kid. My brother today has 2 500s, one in wood stock with a bird barrel and the other in synthetic stock, a slug barrel and a scope for deer hunting and he's had great success with both.
Like I said at the beginning though, you honestly can't go wrong with either the Remington OR the Mossberg. It really just comes down to personal preference. Whichever one you go with, I'd recommend getting one with a 18.5 or 20" barrel for Home Defense, and buy a second barrel of either 26 or 28" for skeet/trap shooting or bird hunting.
As for shell size I say stick with 2.75" shells. I've said it many times on this forum that IMHO 3" shells are overrated and over-hyped by gun and ammo makers. I've hunted literally everything from squirrel to whitetails with my Remington 1100 12 Ga. (which is 2.75" only) and have never found anything I couldn't kill with it because I didn't have 3" shells. My brother tried using 3" for awhile but after finding out about the considerably higher cost and recoil (especially in a pump) he promptly went back to 2.75". Whether hunting game or defending yourself, accuracy and shot placement count for much more than the size/power of the round your using. A 2.75" shell you can shoot accurately and actually hit the target with will be far more effective than a 3" one you can control or shoot well. All the added power of a 3" shell (which isn't that dramatic an increase IMHO) will be worthless if you miss, and yes, you can miss with a shotgun at very close ranges. I've seen it happen! Even for defensive use, good 2.75" buckshot of whatever size you want (#4, 2, 00 etc.) will be more than adequate for home defense, especially when you consider the close distances you probably be shooting within your home and the fact that at those close ranges, say 3-5 yards or less, the pattern will still be relatively small, probably not much bigger than the size of your hand.
Personally I'm not too keen on bullpup shotguns. I say keep it simple and get a gun like the 870 or 500. They're so proven, simple and intuitive for almost anyone to use. No need to reinvent the wheel, especially given how many millions of each have been sold over the past few decades and how many LEOs and military units have depended on them and used them with great success.
Hope that helps.
