For the shotgun, the Mossberg 500, Remington 870, Ithaca and Winchester 1300 are all reliable and low-priced shotguns. I prefer the 12ga, but the 20ga is often better for female shooters due to lower recoil. Your wife should shoot both to see what her recoil tolerance is. A 20 ga is no weakling for a home defense gun. Like others have said, #4 shot in a high-brass duck load would be devestating from across the room, but won’t go through walls with much energy left.
For a handgun there are many more choices. Revolvers are more reliable for people who don’t shoot much or don’t do maintenance on guns much. But, revolvers have more felt recoil because there is just the sudden impulse instead of being spread over the duration of a semi-auto slide operation. Also, on most revolvers there is much more muzzle flip because the hand is so much lower than the barrel than a semi-auto. Hand guns are much harder to give advice on. I would go to a range that rents guns and you and your wife should try to find something you both agree on. Many people say that you should not go smaller than 38 special in revolvers or 9mm in a semi-auto, but I think that some smaller rounds such as the revolver round .327 magnum would be good, plus you can use lower powered 32 rimmed ammo for cheaper practice and less recoil. A downside to something like the 327 magnum is that it is much less common and more expensive than something like 9mm which (usually) can be found easily. Read up on defensive and practice bullet types, but practice enough with the more expensive defense ammo to be sure it works reliably and is accurate in your gun.
I've heard anecdotal evidence lately of all such ammo being snapped up by government agencies (ICE, SSA, etc).