There are several ways to make Gil in FFXIV. Quests, crafting, farming and playing the market board.
Questing- This will get you quite a bit of Gil and progress the story along the way. Just the main scenario quest will net you a lot of Gil but will consume most of the time you have in the game up until end game. This will be your main source of Gil until you're able to do one of the things listed later. Side quests and level quests give a smaller amount but do give you something.
Crafting- The bread and butter of serious FF14 Gil makers. People who take the time and Gil to invest in a crafter are THE richest in all of Korea. I won't lie to you, they put a lot of time, effort and Gil into this. Trust me when I say that this will pay dividends later in the game. HQ items and gear are fantastic early end game gear. If you're serious about making Gil in FFXIV you'll want a crafter.
Farming - Where do you think the crafters get their materials? Probably from farming it themselves. However, those who don't have the time or the patience to do it will buy it all on the market board or from their FC. Don't misunderstand, it is probably one of the easiest ways to make Gil. Just takes time.
Playing the market board - Ok a quote is best suited for this one. “Gotta spend money to make money”. The premise is you gotta find the item that is selling quickly for the least amount of Gil. Buy the item and then sell for a higher amount. If someone has undercut the next highest item by a lot, then they either misclicked or didn't check the price of everything before the put it in the market Final Fantasy 14 Gil board. Either way, your gain. A fair bit of warning. You can seriously mess up the economy of your server if you do this wrong. Granted that's if a lot of people are doing this method wrong.
Bonus - keep an eye out for what materials are needed in the next major patch/ expansion. Those will be the items in most demand and able to get you the richest.