Welcome to the game.
You will select a Grand company as part of the story. This ends up like choosing a Faction in PvP. There is no difference between them, so pick any that you want.
The Free Company is the equivalent of an MMO guild. Usually, these are a group of players with interests in one or more aspects of the game: raiding, housing, etc. Larger and more established FCs will generally have more resources for you to use. You might not actually get to use all these resources straight off, but they have a larger capacity. FCs have different cultures between them, so make sure the FC culture fits you best. It's like a company. You will be spending a lot of time together, so make sure you can get along well with them. The biggest resource for crafters from a FC are the leveling boosts to crafters they can give (if they have the workshop items), and the vendor npcs which give your cheap mats to craft everything up to lv.20.
Take a look at linkshells (LS) for developing your network and friends. These are more like social circles and private channels than guilds. You can be in a maximum of 8 at a time and they can be much more social and likely to teach your= the ropes. Look for a friendly beginner LS first and then join a FC of someone in your LS you think is a good mentor or who you get along with.
Crafters and gatherers are not mandatory at all. If you are interested in leveling them though, you will want to level all crafters to 15 first. The cross-class skills will allow you to create high quality almost all the time (especially once you get carpenter, goldsmith, alchemist, and culinarian). The rest come at levels 38 and 50, but every cross-class skill is useful in crafting items after level 50.
If you go in to farming, I would recommend mining and botanist to be at least level 53. This is to provide you with seeds to plant (botanist) and soil to get fertilizer (mining)that increases yields or cross-pollination to create unique plants only obtainable with farming. You will also need a house for this. You can use your FC house, but other members may be using it and as a new member, you may be squeezed out. Again, it's about the culture.
If you buy your own house, keep in mind the base price on a small house is 20million gil. Larges houses are somewhere around 300million gil, and this was before the expansion. Prices will likely be a lot higher this time around because it's easier to make Gil at level 60. The cost does reduce to half over time, but then you will be competing with a lot of other players over it. Crafters and gatherers are a good way to make money, but you need to be very well geared to make that much money quickly.