I don't know if there is any one correct order of buying shotguns. That said, my own purchases seem to mirror that used by a lot of folks with whom I shoot and hunt.
Oddly enough, the first shotgun I shot was a .410 which was considered a 'boy's gun.' The recoil was certainly light for an elementary school kid, but the small amount of shot in each shell really meant that I'd have to be an experienced shooter to hit with one. In essence, the .410 is a 'man's gun,' but that description was reserved for the shotgun I first bought: a 12 gauge. I chose that shotgun for use on clays, pheasant, turkey, and waterfowl. A few months later I added a second shotgun which was a 20 gauge. The smaller bore was better suited for grouse, woodcock, quail and other smaller birds. Along the way, I added a 16 gauge, a 28 gauge, and even a .410.