Before running out to Swyrgeim, you'll need to pick up a carton of orange juice. You can either buy this from Katherine in Ul'dah - Steps of Thal > Sapphire Avenue Exchange, or craft it from level 6 cooking using three La Noscean Oranges, a fire shard, and a water shard.
Once you have an orange juice in hand, head over to Vesper Bay in Western Thanalan and find Swyrgeim. She will offer you the quest “Color Your World”, which will require you to turn over your orange juice in exchange for ability to use dye.
At this point, you're not limited to just the dye rewarded from the quest. The dye vendors located at the following locations provide a much wider range of choices.
In addition to these vendors, all crafters can synth additional colors at Lv. 30. These colors include: