Not sure if you're confusing the two, but Free Company (FC) is a group of players who join together to help each other, socialize, and get various in-game benefits from that cooperation. Grand Company (GC) is the military arm of one of the three main cities (the Twin Adder of Gridania, the Immortal Flames of Ul'dah, or the Maelstrom of Limsa Lominsa). The main storyline will involve joining a Grand Company, so that part you will do in order to continue through the rest of the game. Joining a Free Company is optional but highly recommended.
If you're interested in them, you can join the ones there in Gridania now and try it out a bit. When you get to the level 15 quests in the main scenario storyline, you'll be sent as an envoy from Gridania to the other main city-states, Ul'dah and Limsa Lominsa. At that point you can join the guilds located there as well.
But you should probably consider this first bit as trying it out to see if you like it (though keeping in mind that it will get more complex as you go up in levels and learn more skills). You'll quickly run out of inventory space if you try to get very far in crafting and gathering before unlocking retainers. Retainers are NPC assistants who do several things for you, including holding extra inventory items and selling stuff for you on the Market Board. They get unlocked by a level 17 Main Scenario quest "Scions of the Seventh Dawn". Once you've gotten that far in the story and hired a couple retainers, then you can get further into the crafting classes.
Crafting in this game is completely optional. Some players like it and focus a lot of their attention there. Others don't ever do any crafting at all. Either works; it's purely a matter of what you enjoy. Those who do get really involved in the crafting system usually end up advancing all eight of the crafting classes and either all three of the gathering classes or at least Botanist and Miner. It helps to do all of them, partially because they provide skills that can be used cross-class (learned by one class and then usable by the others), and partially because they provide materials for each other (like armorer recipes that use some leather provided by leatherworker, or leatherworker recipes that also use some cloth provided by weaver, etc.)
Yeah, there's a lot to do. The things to not lose track of are the Main Scenario Quests (the ones with the flaming ring around the quest icon rather than the Q symbol) which give you the main story and unlock features throughout the rest of the game, and your class quests (which you'll get each 5 levels from the guild hall of every class you unlock). Explore everything else as you please.