> My question is: would it make sense to use my credit card even when I have the money and immediately pay it back?

Yes. That’s how you’re supposed to use credit cards

> From what I understand, the more you use your credit card and pay it back on time, the higher your score gets

Incorrect. Usage has no relation to credit score

Credit Card Basics: https://www.reddit.com/r/CreditCards/wiki/credit_cards_basics

Credit Building: https://www.reddit.com/r/personalfinance/wiki/credit_building