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When you close a credit card you should look at its age (whether or not it will change the average age of your credit), and utilization (whether or not your aggregate utilization change and how significantly). If your utilization will increase significantly, you may want to pay off some of your debt before you close the card. All in all, if you aren't planning to open several new accounts, your score shouldn't change significantly.
As for the credit card, you can apply for a regular credit card with no annual fee. If you plan to use it right after approval, consider credit cards with a 0% intro APR on purchases. A credit card with a bonus offer can also be a good option as you could save on your first purchases using such a card. Ideally, you would want your new credit card to have all - no annual fee, 0% intro APR on purchases, and welcome bonus offer. For example, it can be the Discover it® Miles card. What is great about this card is that you can redeem earned miles as pure cash deposited to your bank account (in addition to other redemption options).