When do you unlock mounts?

[Enkidoh]

That's correct, Grand Companies and Free Companies are two different things - Grand Companies are like the volunteer militaries of each nation made up of adventurers (which the player has to join as part of the storyline), where as Free Companies are like traditional MMO Guilds or clans, run by players. Grand Company membership is necessary for the chocobo companion mount, Free Company membership is not.

As for the other mounts, the ones I'm aware of and how to gain them are (please feel free to clarify or add anything I may have missed!):

Grand Company chocobo: purchase 'chocobo issuance' for 2000 grand company seals from the Grand Company Quartermaster at Grand Company HQ (Grand Company membership needed), then complete the quest 'My Little Chocobo' by trading this item to the chocobo handler in that city (doing this also unlocks the ability to ride all other mounts)
Legacy chocobo: only gained by players of the original version of FFXIV (1.0) who qualified for the Legacy Campaign (this is no longer attainable)
Goobue: could only be gained for a short time in the original version of FFXIV (1.0) (is now unattainable and will never be offered again)
Courel: register the item code for it from the Collector's Edition of ARR
Ahriman: subscribe for 90 days or more of game time
Magitek armour: gained automatically upon completing the current main storyline
Unicorn: quested (requires WHM to be already unlocked)