There are many dyes you can buy from a merchant the cities -- Ul'dah had one and so does Limsa, so I presume Gridania has an NPC that sells dyes as well. I'm not in game now so I can't check on their locations for you, but they are in the NPC merchant areas.

There are also a number of new dyes (including white and black) and it appears the merchants don't sell these, so it seems the only way to get them right now is through crafting.

You first need the appropriate pigment which you can get through gathering. Any crafting class can make the dyes, but the required types of crystals varies depending on the class.

In case you want the location of pigment for Snow White dye...
The Grey Pigment is required for the Snow White dye and you can get it from Level 30 Quarrying nodes in Western La Noscea right where the map narrows as you go between Swiftperch and Aleport.

It also seems to be available from Logging in the South Shroud, but I haven't gotten there yet and don't know the exact location.