Genesis Crystals are the premium currency you'll find in Genshin Impact, and you will only be getting Genesis Crystals from purchases, no other way.
As a result, you only have to worry about these if you purchase them directly, otherwise you can forget they exist.
Primogems can be collected from completing almost any task in the world, opening chests and completing quests included, and these can be traded for Fate items for Wish summons at a rate of 160 Primogems for one Fate.
Primogems can be earned in-game, or you can trade Genesis Crystals for them, at a rate of 1-to-1. Though if you play enough you will easily be able to get Primogems without having to buy Gensis Crystals.
You will only earn Starglitter from Wish summons, and it'll usually be because your summon is a bit disappointing.
You'll most often get Starglitter when your Wish summon includes duplicate characters or weapons, so it's basically a consolation prize. When you have enough, you can trade Starglitter for more Fate.
Stardust is very similar to Starglitter, and can only be acquired through Wish summons. You will get much more of this than Starglitter, and the items you can trade it for are far more varied.
However, the items you can trade Stardust for are limited. You'll only be able to buy 5 of each Fate item with Stardust per month. You can trade it for Mora or other items, but Fate is by far the most worthwhile investment, so if you run out of Fate to buy, keep saving up!
Anemo and Geo Sigils are unique currencies that can only be traded in at shops which sell Ascension materials - you can find these in towns, marked on the map with a diamond symbol.
These sigils can be earned by completing quests and exploring the world near the town. Mondstadt will use Anemo Sigils, while Geo Sigils are used in Liyue. You won't need to worry too much about these. They can also be traded for Mora, if you need, um, more.
Mora is the basic Gold-like currency of the Genshin Impact world. You'll be using this for all of the more basic purchases from shops in the game, like buying ingredients and certain items.
Mora is plentiful, but you will also need Mora in order to use enhancement and level-up items on your weapons and characters, respectively. Spend wisely so you don't run out too early.