Genesis Crystals are a purchase-only currency, that you need to spend money for.
In the Shop tab you can acquire Genesis Crystals in two ways. Note that in the following screenshots, the prices are in AUD (Australian Dollars).
One way is to purchase them directly via Crystal Top-Up. The first time you buy Genesis Crystals you will get twice as many.
The other way is to buy a Blessing of the Welkin Moon which gives you 300 outright, as well as 90 Primogems per day you log in. BEAR IN MIND if you do not log in for a day you will miss out on that day’s Primogems.
You can convert Genesis Crystals directly into Primogems at a 1:1 conversion rate (every 1 Genesis Crystal = 1 Primogem) but in my opinion it isn’t very worth it.
Otherwise you can spend them in the Gift Shop for bundles of items and consumables.
I don’t personally have interest in spending on this game, but I can see why others might. In my opinion the cost of Genesis Crystals is far too high and/or the conversion rate to Primogems is way too low. I’ll be spending $45 for a single roll and likely end up getting all weapons. It’s better to spend for Resin recoveries but even then I don’t play the game constantly, only for an hour or two per day, normally to burn all my Resin and do some quests if I have spare time.
That’s how you get Genesis Crystals in Genshin Impact – by spending on microtransactions.