New to Summoner and returning player, where do I start with getting the new summons?

I've looked on the wiki and guides and it looks like you have to grind out your reputation in order to do some of the trials to get the summons. Where do I start with this? Is there a guide for reputation grinding?

Also where do I need to go to get the sprite summons? I currently only have carbuncle.

The community refers to reputation as Fame. More info, including some quick fame ideas, can be found here.

The sprites are called spirits and are obtained by eating a relevant item called a spirit pact, Light Spirit Pact, for example. The pacts can be obtained from enemies or obtained from other players or the auction house. Some are easier to find and cheaper than others.

Basically the short answer is, you're going to want to collect a buttload of cornettes, to trade to Yoran-Oran in Windurst Walls; as much millioncorn as you can carry, to trade to Melyon in Selbina; and far, far too many yagudo necklaces, to trade to Nanaa Mihgo in Windurst Woods. Those are the usual go-to repeatable quests to grind reputation in Windurst, Bastok and San d'Oria, and Norg respectively. The cornettes and millioncorn can be bought from vendors, but the necklaces are going to come from either the auction house or fresh off the necks of yagudo beastmen.

Spirits are all obtained by using scrolls (Light Spirit Pact, etc.), but are essentially only useful to you for restoring MP using your Elemental Siphon job ability. They can be obtained from defeating certain monsters, but it's just a lot easier to buy them on the auction house.

Your avatars, the actual useless pets, mostly require a series of quests: Trial by (element). In Southern San d'Oria, Port Bastok, Rabao, Norg, Kazham, and Mhaura, you'll see a pair of NPCs, one a tarutaru in a distinctive hooded cloak and the other a member of one of the tall races. Once your reputation in the relevant area is high enough, the tarutaru will tell you about their encounter with one of the prime avatars, and give you a key item that will let you go fight it. After that, you'll need to make your way to the relevant protocrystal.

Trial by Wind requires you to visit the Cloister of Gales in Vollbow, to fight Garuda.
Trial by Water requires you to visit the Cloister of Tides in the Den of Rancor, to fight Leviathan.
Trial by Fire requires you to visit the Cloister of Flames in Ifrit's Cauldron, to fight Ifrit.
Trial by Lightning requires you to visit the Cloister of Storms in the Boyahda Tree, to fight Ramuh.
Trial by Earth requires you to visit the Cloister of Tremors in the Quicksand Caves, to fight Titan.
Trial by Ice requires you to visit the Cloister of Frost in Fei'yin, to fight Shiva.

After completing each of these fights, return to the NPC who assigned you the quest to claim your reward; one of the options is a pact with the avatar you defeated, you want to choose that first.

You're going to need to wait at least a day and visit each of these NPCs again to repeat the fights, because there will also be an item specific to each which is required to obtain the Evoker's Ring, a piece you will end up using throughout your entire career as a summoner.

And then you need to fight them all again, after waiting another day, without returning to the NPC to claim a reward this time. Speak with Leepe-Hoppe in Windurst Waters, and you will be asked for the key items you obtained from the avatar fights (which are lost when claiming rewards.) In exchange, you'll be given the key item required to fight Fenrir, located in the Full Moon Fountain, accessed from Toraimarai Canal. Win the fight, return to Leepe-Hoppe, claim your reward... then repeat that whole process again because this quest also controls one of the Evoker Ring items.

So, in total, you're fighting Ifrit, Leviathan, Titan, Ramuh, Garuda, and Shiva four times each and Fenrir twice, to get your basic summons and an important piece of gear.

Then there's Diabolos, Cait Sith, Siren, Odin, and Alexander, who all require you to progress through a storyline to fight them, but those are a tale for another day.

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Your avatars, the actual useless pets,

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what?

There are also alternatives to the regular avatar quests, called "Trial size trial by ___," which have a lower fame requirement and are intended to be done solo at a lower level. The fights aren't easy for a new player though. Beating them requires that you learn essential summoner soloing skils, mainly kiting (Summon pet, sic on enemy, run away, wait for pet to die, and re-summon before the enemy reaches you).

The regular "high" level fights are easier as they can be demolished in seconds with high level players, but you will have to go through the fame grinding for longer.

After the main 6 elemental avatars, there are a number of others, all of which require story progression:
Alexander and Odin require progress through the Treasures of Aht Urghan storyline (These are really just special summons for your ultimate ability, astral flow)
Atomos and Cait Sith require progress through the Wings of the Goddess storyline
Siren requires completion of the Seekers of Audolin storyline.

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Originally Posted by Alhanelem

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Originally Posted by Zenion

Your avatars, the actual useless pets,

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what?

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Hah, missed that, I meant to say useful pets. Although, I don't know, some days...