FFXIV New player First question

So Ive played so many other MMO's and though this was going to be similar, boy was I wrong. This game is different in many ways from other MMO's and im enjoying it so much more.

Ive yet to start messing with other jobs/classes but I have a question about that.

How does gathering work for crafting/gathering professions wont your inventory get filled up? In other games you have a bank with many slots is there something similar here so U can horde things you need?

The quests ive taken so far have been very help full getting me to understand the game but I did not see a helper on this one or I may have missed it.

Unfortunately I don't do much crafting and I've never bothered with gathering so I can't really answer your first question, but for your second question about inventory space, as you play through the storyline around level 17 you would have joined the Scions of the Seventh Dawn, during which would have unlocked the Retainer system. 

Although Retainers are specifically used for the market function, thus allowing you to sell FFXIV Items on the Market Board (MB), a secondary function Retainers have is that they also function as item storage for you, meaning you have somewhere to store stuff once your inventory gets too clogged (and it will). By default you can hire two retainers for free, but you can actually purchase more (up to three more I think) through the Mogstation website, which is an extra monthly charge to your standard subscription for the game. But this is completely optional.

Once you unlock retainers to hire them speak with a npc who has a title under their name of 'retainer vocate' (they're found in the market of each city, and are marked on the map with a special 'two faces' icon). When you hire a retainer, you get to choose their race, gender, facial appearance, name etc, just like when you created your character. Once this is done, to actually use them, find a summoning bell in each town or city and select 'Summon Retainer'. It's really simple once you get the hang of it.

Retainers serve multiple purposes. They are your bank; each one comes with 175 inventory slots. They enable you to put things up for sale on the player market (if you wondered where the stuff for sale on the Market Board comes from, those are other people's retainers). Each retainer comes with 20 slots for selling ffxiv gil items. Once you've hired retainers, you'll get a side quest to unlock ventures, which are a way to dispatch retainers to get loot for you. You'll assign the retainer a combat or gathering class and equip it appropriately. It can then be levelled up through ventures to get you better loot. To send a retainer on a venture, you'll have to pay it in "Ventures", which are an inventory-stored currency that you can get from various quests or purchase for company seals at your Grand Company store (you'll join one of the three Grand Companies a little further down the storyline). The venture will take a certain amount of real time (depending on what kind of venture you sent the retainer on) and must be manually collected when it's done. The retainer remains completely available for its other functions while on a venture.

You're allowed two retainers; they cost you nothing except for paying for ventures. You can also purchase an allowance for up to two more retainers by incurring an additional charge of $2/month per additional retainer at the Mog Stations. You hire them by talking to the Retainer Vocates at the main merchant areas in the three main cities; you'll need to log on again to get it to recognize you've purchased addional retainer allowances at the Mog Stations.