A new federal credit-card law that takes effect Monday could erase billions of dollars a year in fees and interest charges paid by consumers. But card issuers are already deploying new tactics that could prove costly for even the most cautious cardholder.

The law made some important changes. Card companies must now tell customers how long it would take to pay off the balance if they only make the minimum monthly payment. Customers can only exceed their credit limit if they agree ahead of time to pay a penalty fee. And unless...