Low-budget can mean a lot of different things, so perhaps the next datapoint to capture is the amount of money that y'all can spend on the gun, and then the budget for range time/ammo. I always recommend that folk not spend all their cash on the gun and leave no money for recurring practice - running a shotgun effectively requires that you have physical familiarity with it, and that means range fees and ammo costs.
If the new-gun budget is less than $300, you can effectively rule out any semiauto worth having, barring a minor miracle. Of the choices in new pump guns, you'd be hard pressed to find a bad one so long as you shop for one that fits and appeals to your bud. I'm an 870 guy, but I have owned and liked Mossbergs as well. The 870 simply fits my hands better, and so I prefer it. It sounds weird, that guns have 'feel', but they do - and that 'feel' is very important. The shotgun is pointed and not aimed, and so it has to move like an extension of the body if you want to be effective with it....
12ga is the standard for HD, simply because it works and a variety of loads are available for it. A 12ga loaded with low-recoil (law enforcement) buckshot and recoil is very very tolerable, even for my fifteen year old kid. If you go 20ga, you will have fewer home defense (HD) or self-defense (SD) loads available but you can get #3 buck loads from a variety of manufacturers.
