Okay, FFXI is the online version right? so, That means that are only:

Carbuncle(starter Summon)

Ramuh
Leviathan
Shiva
Garuda
Titan
Ifrit
(these you have to fight in order to gain their allegiance)

When you defeat a summon, primeavatar, you get to pick from a list of rewards at the NPC that gave you the tuning fork. Items range from element resistant equipment or damage causing weapons, gil, and a 'pact' with the avatar so you can summon it. These summons can be fought again and again, it's a repeatable quest.

Sorry, there is no Bahamut for all of you fans! However, with the recent updates, there is a Fenrir, the newest summon.

Btw, to fight the summons, you need an extremely good amount of level of fame in the areas the tuning forks can be obtained!

I'm in midgardsormr server, I have them aaaaallll! <3 except Fenrir. ^ ^

So, do repeatable quests, lots of unique quests, and talk to the tarus that say "Owzie, don't make me talk or I hurt all over." (something like that!) They are the ones that give the avatar quests. They can be found in North San d'Oria, Kazham, Mhaura, Port Bastok, Rabao, and Norg.

Have fun! I miss you guys!