So, before you apply for a new credit card, find out what your credit score is. Unfortunately, credit scores are not available for free like credit reports are. Try www.myfico.com to learn your true FICO score, developed by industry pioneer Fair Isaac.
If your score is in the high 700s, a few points lost because of a credit application won't matter. But if it's lower than that, every point matters. So, identify several credit cards you are interested in and research what their qualifications are. Then, once you have found one you believe you will be approved for, apply. If and only if you are rejected, apply for another.
Here are some resources to help you find low interest credit cards:
The Federal Reserve surveys credit card companies and publishes their interest rates, etc.
Card Trak is a consumer advocacy group that tracks credit card rates and fees.
Creditcards.com profiles scores of low interest cards.