Between 1 and 15, you can get mats for cheap from a vendor in each of the starting cities, usually labeled "Tradecraft Supplier". Craft various items from the crafting log; each time you have crafted an item for the first time, you get bonus exp for doing it. Shouldn't take too long to get to 15.
If you have ranked up in GC, there's also an item "Company [something]" that looks like an open yellow scroll that gives you bonus exp for crafting while it is up as a buff.
Another slightly helpful thing is that in the FC house there are various crafting furniture (on the first floor I think) that if you use it under level 40, will grant you a bit of extra CP to use.
Eating food of any kind will also grab you +3% exp as you craft, if you buy mineral water or some other really cheap food that can help you over the course of 60 levels obviously.
When you hit level 15, each crafting class gets a unique ability. You can cross class those abilities to help you with leveling the others. Culinarian is usually the first you should level because of
Hasty Touch (lvl 15) and
Steady Hand II (lvl 37 (lol)). For 'low level' leveling, Carpenter's
Rumination (lvl 15) might be a bit helpful, as well as Alchemist's
Tricks of the Trade (lvl 15). The point of obtaining all these abilities is to make it easier for you craft HQ items.
The crafting class quests usually give equipment as rewards every 5 levels, so you can mish mash gear from doing them. You generally don't need to buy gear except for accessories, which influence how much CP you have.
Each day, your GC will have 'Hand In' quests where you make a specified item (the item is generated depending on your class level) and then you hand it in at your GC HQ or to the GC hand in guys around the world. If you hand in an HQ item, it will double your exp reward. If the item has a star next to it, the exp reward will be greater. So ideally you hand in an HQ item to a starred mark for the day.
For levequests,
use this to check which ones have the highest exp gain and easiest stuff to make for your level. Sometimes you could keep doing the levequests below your level for a bit because the items to make for the next batch of levequests are still difficult. There are quests that are "triple" quests, which means you can hand in items up to 3 times before the levequest can be closed, and you get exp each time; however, you have to craft a lot, so you will need a lot of mats. The missions that are probably the least trouble and the most exp gain (if you HQ the item) are the "delivery" quests, where you have to travel all the way out to nowhere to hand in the item. But the teleport fee is not much considering you usually get a gil reward too.
For post-50 leveling, I would say you should have Carpenter's
Byregot's Blessing (lvl 50) and Weaver's
Careful Synthesis II (lvl 50).
The way to HQ at low level is to use HQ mats and YOLO. I leveled my gathering classes at around or ahead of my crafting classes, so I was able to supply myself with HQ mats. Culinarian was actually somewhat simple to level because most of its mats came from Botanist gathering. If that's not feasible or not your style, it's okay to buy off marketboard, or barter with FC mates.
The YOLO part is where I abused Hasty Touch, Steady Hands II, Tricks of the Trade, and Goldsmith's
Manipulation (lvl 15) for everything to up the quality of low level items before finishing it. The YOLO comes from Hasty Touch and Armorer's
Rapid Synthesis, so I guess in the end I used a lot of different abilities lol
If you want the semi-rotation I used you can ask, this post is long enough already