Oooh nice. I have been thinking about trying summoner so that's good to hear.
In that case I might as well elaborate on how these things are connected: like I mentioned, there's a reputation system associated with quests - each city assigns you a reputation rating that increases as you complete quests for that city (and some closely related ones, like Selbina for Bastok and Mhaura for Windurst, IIRC), with the reputation of all starter cities contributing to Jeuno (hub city, at the center of the game's geographical 3-way simmetry) reputation - you want to raise this as it'll unlock quests to increase your inventory (look up "gobbiebag").
Like I mentioned, there's an organization called Tenshodo with a few representatives in all the main cities, and doing quests for them increases reputation with them, but their "capital" is in a smaller city called Norg, inside a cave in the Elshimo lowlands - getting there will involve a quest to be able to access the airship to that island (different from the requirements to access airships between the starter cities and Jeuno) and navigating a labyrintic jungle.
Norg quests run entirely on Tenshodo reputation, and you'll need level 4 reputation to access the quest to get Leviathan as an avatar for summoner, IIRC.
All the other avatar quests can be accessed in different cities (I think you need some reputation to access those, so questing is important), most of which are easier to reach than Norg - the NPCs that handle them are usually a taru along with a member of a taller race. The taru can give the quest to beat the Prime version of the avatar to adquire it on any job, while the taller NPC will only give a quest if you're a summoner at level 20 or above - that specific version of the quest involves fighting a special avatar battle at level 20 (I think you're level-capped for that, I haven't done that version), but can teleport you to the location of the battle, but with the constraints on that version,, it may require some considerable preparation.
Personally, I've always preferred to do as many different quests as possible (I like to see the stories, and there are plenty of connections between a lot of them when you lest expect them), which ends up raising reputation neatly over time, but there are also quite a few repeatable ones, which I don't think add up as much per repetition, but'll help you reach any specific access goals you may have, the wiki has a quick reputation guide which covers practically all areas you might need.
Ah, to look back and remember I unlocked the bard and beastmaster jobs to raise Jeuno reputation for the gobbiebag quests...
For Tenshodo/Norg reputation, once I had airship access, what I resorted to a lot was the smuggling quest from San d'Oria, which took no toll on my inventory (unlike the zinc ore one from Bastok and necklaces one at Windurst) and got me 1000 gil every time I succeeded, though that requires some main mission progress.