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 have been late less than three times in the past year, and the payments were no more than 30 days late, you probably have a pretty good chance at getting a home loan at a competitive interest rate. Lenders guidelines will vary, but most lenders will excuse a couple of minor 'late-pays' as long as the borrower can provide a reasonable excuse explaining them (i.e. job transition, illness). If the late-pays were 60+ days late and cannot be explained, you may have to settle for a higher interest rate.