How to catch a Dragon-type in Pokemon GO is a lot more difficult than it sounds.
Dragon-types are some of the most fierce Pokemon trainers can add to their battle team. They were, for quite some time, known as the apex of mastery within the Pokemon community. Many legendary Pokemon are dragon-types and several of them are known to be fierce and rare contenders. This trend has only continued in Pokemon GO with the hype of Dragonite around the game's launch almost four years ago.
How to Catch a Dragon Type in Pokemon GO
Your first step is to, of course, familiarize yourself with exactly what dragon-type you want to catch. Some can be found simply by hatching eggs while others need to be hunted. For example, trainers can hatch Axew from 10km eggs, but they'll need to actively search for others.
It may also be worth your while to read-up on any changes noted with egg hatching. It's always possible that Niantic Labs may switch the Pokemon available to be hatched between patches.
It should also be noted that not all dragon-types are "born dragons." For example, Swablu is a normal and flying-type that evolves into Altaria, which is a dragon and flying-type. This also goes for Trapinch which eventually becomes Flygon.
A dragon's secondary-typing often betrays the best methods to finding it. Pokemon like Gible and its evolutions, Gabite and Garchomp, may be easier found where ground-types spawn. Deino, with its dark-typing, may be more common at night.
Dragon-types are also popularly featured in raids and other similar events.
Beyond encountering dragons, you'll need to actually catch them. Ultra balls from PokeStops make this easier, as do berries such as regular and Golden Razz berries. Nanab berries may be useful to keep flightier dragons in place.