I'm at Goldsmithing 41 and not doing too badly at not going bankrupt so far

. Here are some of my handy hints and tips.
I'd recommend getting
Moghancement:Desynthesis if you're planning on going down the desynth route. The success rate will still be bad compared to synthesis, but it will reduce your chances of losing your ingredients when you break.
It's worth keeping some of those Brass Flowerpots and growing your own
Fire Crystals and
Lightning Crystals if you're not into gardening already. Keep any coloured stones you get for synthing into gems.
Keep an eye open for bargains at all the Auction Houses. Buy Zinc/Silver Ores for cheap (e.g. 100g for silver and up to 1000g for zinc). Mine when you can, and don't discount the usefulness of Zehrun Mines for ores. Remember that Bastok is usually more expensive than Jeuno. San d'Oria often has cheap ores and Windurst is usually the most expensive with poor choice.
Make your own ingots, and make double what you need, then sell the rest on the AH.
Find yourself a niche, or preferably several, of in demand items that you can sell for a profit. They are out there, but you'll have to find them yourself. Pretty much all of what I'm telling you is common knowledge for people who've been crafting for a while, so I'm not giving anything away. However, beyond the hint above about the ingots, you'll have to do your own market research.
You can buy
Silver Earrings at the jewellers in Jeuno. Desynth them when Silver Ore is expensive. 315g for Silver Ore is your friend - I'm currently at the stage of grabbing every Mythril Ore at 1000g I can find, and it's about to get much, much more expensive. Really, 1260g+crystal per synth is incredibly cheap.
Check the goldsmithing outlets when the main shop gets expensive. The more of an item that's sold, the more expensive it will be the next day (or it just sells out). The stock resets after the guild holiday (Iceday for goldsmithing).
Check the daily guild point item lists. You may be able to make some quick cash selling items that will only be in demand for one day. Similarly, you may be able to combine your skilling up with items you can turn in for points yourself (once you get to 28).
Don't be afraid of paying for advance synthesis support.
Take your time. Sometimes everything goes right, sometimes you break every synth and never see a skill up. Some people swear by
synthing at a particular day/time/moon phase/direction. Others claim this is just superstition. Ask 100 crafters how skilling up/Hqing/not breaking works and you'll get 533 different answers and a severe headache.