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

Not necessarily, if you've been late with your payments less than 3-times in the past year, and the payments were no more than 30-days late, you still have a good change at getting a competitive interest rate. Most lenders will accept certain reasons for this like an illness, or job-change, but explanations are required.