You don't spend much time on the range, do you? It's easy to miss with a shotgun, particularly if the target is more than a few feet away.
USE the sights and AIM. Get a ghost-ring rear sight
" For home self defense, nothing beats a folding stock, short barreled shotgun.
You don't have to aim it, just point and anything in front of you will die. "
How many gun myths can you count in the above two sentences?
1. Nothing beats a shotgun
(Yes, the ar15 platform does, for shear number of rounds, reduced risk of overpenetration, range if needed, etc- 30 rounds of 5.56 with a fast followup, and the general "dumb people think it's a machiine gun" factor)
2. Foding stock shotgun
(disliked by many because of discomfort with actual use)
3. You don't have to aim it, just point and anything in front of you will die
(giving advice like that will get somebody killed - likely the person being told they didn't have to aim to take on their target. Any gun must be aimed - a shotgun doesn't pattern based on magic, there are many factors involved.)
You still have to practice. Anybody who does skeet, trap, or bird hunting will tell you that you don't aim it, you point it . . . but you still have to practice pointing.
Any goblins that bother you are probably going to be closer than a skeet or trap clay and a good deal larger . . . but you can't just point the shotgun in their general direction and hope for the best. Practice, practice, practice! I concur that the Remington 870 is your best choice . . . and if you ever get into bird hunting (or hunt tests) you have got yourself the perfect all purpose bird gun too. 12 ga has a pretty hefty kick so you need to be ready for that, if that worries you, get a 20 ga. or a 16 ga. if you can find it. My dad shot a burglar with a 16 ga. double (with bird shot) and the guy never came back . . .
I would also recommend a reliable sidearm for when you can't get at your shotgun. Since you are starting out, stay away from the autoloaders for a first gun. I would recommend a good .357 revolver with a short but not stubby barrel (stubbies lose too much velocity and are hard to aim, plus they kick worse.) The great thing about the revolver is it is the original point and click . . . no slide or jams or magazine to worry about. You'll only get six (or five) shots, but you can buy a couple of speedloaders and (again) practice, practice, practice. The great thing about .357 is it holds .38 Specials too, so you can practice, practice, practice with the cheaper (and low recoil) .38s until you are confident.
If you really get into shooting is time enough to think about getting a semi-auto . . . I swear by my 1911A1 and .45 ACP is a better caliber than .357, but you don't want to fool with all the extra stuff to learn on your first handgun.
Just my