How to earn more Acquiant Fate and Intertwined Fate and why
Acquiant Fate and Intertwined Fate are both the currency types you'll need to summon more characters from the "Wish" menu, and while you might get a bunch to start with, you'll find that continuing to earn more or it can be quite difficult.
Genshin Impact's development team MiHoYo actually specifically recommend that players don't throw away their newfound premium currency too hastily - you should save it and spend it wisely.
In this case, you want to use 10x summons almost exclusively. 10x summons give you guaranteed chances of earning new characters and high-level weapons, and if you spend 10 summons separately, you will get far worse results. Save up, and spend wisely.
Beginner's Wish is the best Wish method for brand new players, and you should entirely avoid all other Wish/Summon types until you have summoned a total of 20 times from Beginner's Wish, and yes, that should be two 10x summons.
Your first 10x Beginner's Wish summon guarantees you earn Noelle, a four-star character who uses clubs and greatswords. She looks cool, and plays great, so she's a good option.
Your second 10x Beginner's Wish summon guarantees yet another high-level character, though that could be anyone. Regardless, getting guaranteed characters this early in the game is incredibly valuable to your progress long-term.
How to check appearance rates and what you will get
Before you spend your Fate on anything you should know your odds, that is key to anyone compulsive gambling scheme. If you don't know the odds, you're just another shmuck getting played.
So, before spending your fate on any kind of Wish, hit the Details button in the bottom left of the screen. Check here to find which characters are available, how rare they are, etc.
For example, in the Beginner's Wish there are three 5-star characters available, but their appearance rate it 0.6%. Pretty brutal, honestly. 4-star characters are slightly better, at 5.1%, not counting Noelle, but it proves that you should be saving for those 10x summons to get far better chances of getting good characters.
Grinding away for fate
Once all is said and done in the Wish menu, you'll find yourself with less Fate than before. What a surprise, eh? Well, it's time to get it back, and if we can do that without spending actual money, that is a bonus.
The first method of getting extra Fate, and the most direct, is by visiting the Shop menu and checking out Paimon's Bargains. There should be amounts of Fate on sale in exchange for different types of currencies that you can earn while playing the game, and these will make it far easier to summon new characters and weapons.
You can also earn Fate directly from completing quests and achievements as you play the game (make sure to redeem those rewards frequently) but ultimately it will be the Shop which will be the main source of acquiring easy Fate long-term, so keep completing quests, earning currency, and trading it in so you can summon yourself the best party possible.