We helped DS open a credit card account when he was 17 & driving; we did not want him using a debit card to purchase gas. This card was with the same bank as his checking account; due to his age, I was also on the account. The credit limit was deliberately low, $500, no annual fee, and earned 1x cash back rewards. (Later on, this was replaced with a higher earning card.) DS was totally responsible for paying his card in full every month. I know parents who paid their student's card but this does not build financial literacy or responsibility. Because I was on the account, I could & did monitor it online, & DS knew that I would.
DS added a second card while in college: the CapitalOne Quicksilver card, which earns 1.5x cash back, no annual fee, & no foreign transaction fees. He is now 25 & his credit score is in the high 700s; he has never missed a payment or carried a balance. He uses the Citibank Double Cash Mastercard for most purchases, earning 2x cash back, no annual fee, but it does charge a foreign transaction fee. Recently he added the Barclays Uber Visa card, which earns 4x cash back for dining, 3x for travel, 2x for online purchases, 1x for all else, no annual fee, no foreign transaction fees.