It's no secret that Genshin Impact's gacha system is designed to make you want to spend money. You absolutely don't need to do this, though. Since launch, I've been enjoying the game without spending a dime (I should also say I'm not part of any of miHoYo's content creator programs that deal out free currency to content creators).
I've managed to do this all while building out a roster with all the characters I want. How, you might ask? Well, the short answer is math, a pessimistic attitude, and discerning taste. You can also read on below to understand exactly what I mean by this.
The math is bad
It's probably not a shock to hear that Genshin Impact's virtual slot machine is tuned to make it extremely unlikely that you'll get what you want out of it. To be precise, you have a 0.6% chance of earning a coveted five-star adventurer or weapon on any given pull. To see what that looks like, see the video above where I burn through a ton of Primogems to (eventually) get Hu Tao, the game's latest character.
You may see screenshots on Reddit or Discord of players pulling multiple five-stars at once, and it may convince you that the odds aren't that bad, but this would be a mistake. These online communities are full of hundreds of thousands of players, and with all of them pulling regularly, you're bound to run across individual stories of people getting very, very lucky.
Pessimism is your friend
The best way to go about using the gacha is to assume you'll never get one of those lucky pulls. In fact, it's best to assume that the only way you'll get a five-star character or item is through the game's "pity" system, which guarantees a a five-star drop after 90 pulls without one.
This is to say that you shouldn't pull for a character unless you have at least 90 pulls saved up. If you want to be extra careful (which you should be), you should actually save up 180 pulls worth of Primogems, as Genshin Impact's pity system only gives you a 50% chance of receiving a rate-up character when it activates. You have an equal chance of pulling a standard five-star character.
The only exception to this rule is if the last five-star character you've pulled from an event banner wasn't the rate-up character. In this instance, your next five-star is guaranteed to be the rate-up character.
The final tip on managing your Primogems as a free player is to be incredibly choosy with which characters you go after. This is your hard-earned currency! Don't go and toss it away on a character you think is kinda cool. Spend it on characters you deem must-have!
If you aren't looking at a banner with a character you love, don't pull. At all. Not even to "up your pity rate counter" or whatever. The rates may be bad, and you may be assuming you'll only get a drop on pull number 90, but you never know.
This is how you can save to actually be able to guarantee characters you want. I hadn't spent a Primogem since November after getting Childe, which allowed me to bank them to ensure I'd get Hu Tao. Yes, this means I missed out on the chance to get characters like Zhongli, Albedo, and Xiao, but that doesn't matter to me. I have two characters I really like, and I didn't have to "hope" I'd get them. I knew I would.