EDIT: You need to reach lvl 30 to even GO on the summoner quest with your main job. Then you get summoner and poof, back to lvl 1. ^_^

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First you have to get Carbuncle, gather the lights in the Red Ruby all over the world, long and soo annoying. The other Avatars, you have to go on quests and fight them to have the priviledge to call on them, kinda like how in FFVIII you had to challenge Ifrit :/ You have to reach Summoner first but I think you can start gathering the elementals around but they're quite tough to gather. You can't challenge the avatars until you're level 20 I think in Summoner.