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Groceries cost the same with or without a credit card. The store is paying the card every transaction, not the user. Annual fee cards need more scrutiny and are often not worth it unless you’re playing the travel points game (and what a game it is). No card in existence is worth it unless you pay it off in full every month.
They even have non-cashback/points perks that are worthwhile, like rental car/travel/phone insurance. My family has saved hundreds with free phone insurance from a card. My screen stopped working, and I got a new one for $30 from an authorized repair joint, which otherwise would have cost like $300.
So, yes, cards do give you free money so long as you don’t hold a balance.