Consider Capital One® Platinum Credit Card, the card has no annual fee for the first year. Your credit limit will depend on your creditworthiness. If you want to increase your credit limit, you should use the card responsibly and pay your bill on time. After six months of card use you can request a credit limit increase. You will have to use the card to prove your ability to pay and to show a need for the increase.
You can apply for any other credit card you like on our fair credit page. These cards may not offer automatic credit increase, but you can always request one. If you have acted responsibly with the credit card, the issuer will generally increase your credit limit.