I've only been late a couple of times on my credit card bills. Does this mean I will have to pay an extremely high interest rate?

Not necessarily, if you've been late with your payments less than three times in the past year, and the payments were no more than 30-days late, you still have a good chance at getting a competitive interest rate. Most lenders will accept certain reasons for this such as an illness or job change, but explanations are required.