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I got a bank of america credit card at 17
Capital One gave me my first card (Quicksilver One) starting out with a $300 limit and it really helped me build. They gave me an automatic credit limit increase after 6 months. After that I got a couple store cards and made regular purchases and payments on them and then was finally able to get approved for Barclay Rewards, Chase Freedom, and Amex Blue Cash Everyday.
My husband's first card was Capital One Platinum with a starting limit of $500.