Obtaining Crafting Ingredients
After gathering tools, you're going to need actual ingredients to go into these crafts. As mentioned in the Final Fantasy 14: A Realm Reborn classes and jobs guide, Botanists, Fishers, and Miners are strictly meant to gather materials. For other crafts, gathering may not be so straightforward.
Ingredients drop from enemies fairly often, but your best bet is complete guildleves. Don't let the term frighten you - a guildleve is simply a quest given to you by your respective guild. Doing these quests will earn you materials as quest rewards. By the same token, many non-guild quests offer a choice of reward at completion. Often, one of these choices is a material. If crafting is your goal, choose that material. They are usually quite specific and hard to obtain otherwise.
Leveling Up Your Crafting Skills
Leveling your crafting skills can be done three ways. The first is with simple grinding. The more materials you gather and turn into items, the more experience you'll earn. It's simple and it works, but it's boring.
The more enjoyable way to level your skills is again through guildleves. These are actual quests, but on top of that, they offer a lot of experience upon completion.
Finally, you can go through the game's Crafting Log and make items on the list. The more you make, both overall and uniquely, the more the experience will flow. In the same vein, using high level abilities or crafting HQ items will net you much more experience than a typical synthesis. But be aware that both of these options take more resources and have a greater chance of failure.