1. Have Miner and Goldsmith unlocked. In order to do this you must do a Level 10 quest of any of your Disciple of War/Magic job of your choice. Unlocking Weaving and Armorer is recommended but not required - the only advantages of those jobs is the quest to get some free crafting armor. Weaving has another advantage: you can eventually craft your own crafting gear that you cannot get in the shops.
2. Level up mining first. If you're going to Goldsmithing the focus should be on Copper Ores, Zinc Ores, Bone Femurs, Raw Sunstones, Raw Malachite, etc. If you're going to work on Armorer, it should be on Copper and Tin Ores. Do not sell any of the items you mined to the NPC. Depending on what you are crafting, you will need at least 300-500 of certain items. Mining the items will save you money as getting massive amounts of gil is hard to do in the early game.
3. Get as many Wind and Fire Shards that you can. Killing the elementals, finishing quests, and mining can get you. Those stack to 9999 and you will need a lot of them, at least 1000 to get to Level 1-20 in any one craft.
4. Buy food at the shop near the Black Brush Station. One food increases mining/botany (Rabbit), the other increases crafting. Do not craft without any of those food effects active, as the extra 3% XP boost will add up fast. The food of choice gives boost to craftsmanship and control and it lasts 30 minutes. Buy 99 of them.
5. Try to power level on consumables such as Ingots, Sunstones, etc. Avoid armor/weapons except to complete the crafting log or when its needed for a quest.
Mining from 1-10 should be focused on Copper Ores, Tin Ores, and Zinc Ores. Mining from 10-20 should be focused on Zinc Ores and Raw Sunstones. Only mine the other items to fulfill Gathering Logs and Guild Quests.
Goldsmithing 1-8 should be focused on Copper Ingots. Craft the other items only for Crafting Log purposes one time unless the item is needed for a quest.
Goldsmithing 9-17 should be focused on Sunstones and Brass Ingots.