Carpentry is located in New Gridania. Speak with the Guildmaster, you can find them on the minimap under the Carpenters Guild. They will give you a quest that teaches you how to synthesize your first item.
You can purchase more supplies from the Guild Supplier merchant in the same room as this if needed.
You must synthesize 1000 total items under the Carpentry trade. You can do this naturally by leveling up and progressing down your item tree, or pick one cheap item to make over and over 1000 times total.