Welcome to Eorzea!

Firstly, you are correct in that normal story quests/side quests are 'once-only' - once it's completed, you can't do it again. However, don't despair, as there are other alternatives to sidequests for levelling:

Levequests (which are repeatable as long as you have enough allowances which recharge every six hours)
FATEs (still the easiest really, as you can do as many as you like and come and go from them at will)
Dungeons (if you do them via the Duty Roulette you can get a bonus to exp and gil once a day)
Hunting Log (each class has it's own log unlocked automatically upon completing it's first class quest for it's guild)
Challenge Log (requires a quest to be completed to be unlocked - this is kind of like a cross between achievements and the Hunting Log. Completing various objectives, such as, complete x number of levequests from five different areas, or gain the highest rank in a FATE x number of times, will give you large amounts of gil and exp. This resets once a week on a Tuesday)
Beastmen 'daily' quests (unlockable after around level 40, each beastman tribe has a 'friendly' faction who will have three quests each day for you to do - you're restricted to only being able to do three per tribe for a maximum of six per day, but they too reset each day at Japanese midnight. Npcs with beastmen quests can be determined by the speech bubble icon with a 'recycling' symbol)

Additionally, there is an added effect you're given if you're levelling a class lower than your first in that you're given a bonus to exp gained after battle for any class that has a lower level than your first (so for instance, your current class is ARC 28, all other classes will gain an exp bonus up to level 28). Not to mention logging out in a 'sanctuary' (either of the three cities, or a hamlet which has an aetheryte) will also grant you a 'rested exp bonus' on top of that. I found levelling was far easier with later classes than it was the first time, mostly due to this exp bonus.

Secondly, there is actually no true 'Auction House' in FFXIV. Instead, we have what is called the 'Market Board'. This is basically just a list of player retainers who are virtually personal shops. When you purchase an item on the Market Board you're buying that item from a player directly and not actually bidding on it. To sell items yourself you first have to hire a retainer from a special npc in the Market area of each city - this npc has the title 'Retainer Vocate' under their name. They're shown on the map as an icon with two heads. When you hire a retainer, that person becomes not only a shop front on the Market Board for you, they also give you extra storage space for items (this is completely seperate from the Market). To use a retainer after hiring, simply look for a 'summoning bell' in various places (one is in your inn room for instance). By default, you can have up to two retainers, but you can purchase more for real money from the Mogstation, although personally I find having two is enough.

Anyway, hopefully that hasn't been too confusing for you! FFXIV is full of references to past FF games so if you';re interested in all the in-jokes, lore and references, the FF Wiki is a good place to get up to speed.