Modern times in the USA? Remington make a 'Youth Model 11-87' variant of their venerable semi-automatic shotgun. Ditto for their pump action Model 870. These are guns with shortened stocks and barrels, chambered in 20 gauge.
Age that a fit kid could swing one? 12 to 14, being realistic for a boy, perhaps 14 to 16 for a girl.
Below that age a .410 gauge coach gun, that is a short barreled, side by side double barrel shotgun, would be the easiest and therefore safest for them to handle, especially as a'house gun'. Exposed hammers would be out. Just one more thing to slip or catch on.
For hunting then a .410 with full length barrels would be a possibility, these were once common, but have long disappeared from most catalogues.
The .410 is sometimes called a 'snake charmer', for very short range disposal of snakes in the farm yard. The limited amount of shot in the cartridge means the pattern becomes dispersed quickly as distance increases, makng them ineffective for wing shooting birds or running rabbits.
if you are willing to consider other options, author Michael Z Williamson rather famously provided his pre-teen daughter with an AR15 with collapsible stock as her first gun. Low recoil, relatively light weight, and length of pull from butt to trigger was adjustible for her height and reach as she grew up.
http://blog.olegvolk.net/2013/07/01/they-grow-up-so-fast/
A bolt action .22 rifle is usually considered the starting point for kids with guns, shooting with scouts, small bore rifle clubs or at side show alley(many years ago). Currently there is a minature rifle for preteens called the Cricket and a few of the better quality manufacturers such as CZ make cut down versions of their adult rifles for early teens.
The Bond Arms derringer mentioned by Ernest is heavy, has rudimentary sights and a truly awful trigger. I doubt a child could pull it without using both index fingers.
If the story setting is in the period from the 1880s to 1950s, then the most commonly found firearm would be a single shot break action shotgun, such as the Harrington & Richardson Topper, which has been manufactured since the 1890s.