When I joined Final Fantasy XIV I joined with a few of my real life friends and some old friends from Final Fantasy XI. We decided we would all pick a different main craft so we could make and trade gear rather than level overlapping crafts. It's working out really well.
The two I picked were cooking and weaver. Weaver is my highest so far and I've heard a lot of people say you can't make money with it or any craft. No way! I make plenty of gil with my weaver. It's just not as simple as make and sell gear. You have to think a bit more carefully.
So, here are my top five tips.
Level botany along with weaver. Your main ingredients will be moko grass (really cheap) > cotton bolls (1k a stack from the guild merchant) > diremite webs (farmed from monsters) > flax (14k a stack on the marketboard or harvested) > fleece (from monsters). You can skip those diremite levels by harvesting cotton and making lots of cotton stuff. It'll be easier and cheaper since diremite webs are 200-400g a piece or farmed from a very over camped area. Ideally keep botany five levels higher than weaver.
Make friends with a goldsmith and blacksmith/armorer as well as a leathercrafter if you can. Some things will take an ingot here or there or a leather and it's so much easier to be able to have a fellow crafter make the than buy them on the market board.
Don't worry about finishing your crafting log if you are trying to make money. It'll be cheaper to focus on making thread and then cloth. I sell the thread and cloth on the market board unless I see armor that sells well. The crafting gear sometimes sells well on the market board. When you get new recipes take a look and see if any is selling for a decent enough price. If it is, don't make more than 3 at a time because prices change quickly.
Make your own crafting gear. Always upgrade when you have a higher level piece of crafting gear you can make. You'll be able to make a lot of crafting and gathering gear. I had a leathercraft and goldsmith friend and I would trade the cloth pieces for what they could make. If you can't do that, it IS worth buying the gear but don't worry about buying the HQ unless you have gil to spare.
Unless you have something specific you are saving your crafting gear for, when it's time to upgrade to new gear you should convert old gear to materia and either use that materia on new gear or sell it.
On quests where you need to turn in an HQ item to your guild, make sure to check if it would be cheaper to buy the item than try two or three times to get an HQ.
Even when you get high enough to make a lot of good gear, better profit seems to come from thread and cloth because other crafters use it.
NEVER USE QUICK SYNTH unless you have to. You get more exp plus high quality threads mean high quality cloth which increases your chance of high quality gear.
Even cheap control food from the food merchant helps. Use it for more exp.
Want to make gil without weaver?
How to Earn Gil in Final Fantasy XIV Part 1 - http://www.bubblews.com/news/1112789-how-to-earn-gil-in-final-fantasy-14-part-1-shards-and-a-bonus-method