Bad sectors usually indicate a dying drive. Formatting and reinstalling the OS will only allow files to know where the current ones are but does not protect the system from more sectors going bad (which most likely the will). As such, if you want to ensure that you don't lose your data it would probably be in your best interest to make a purchase of a new one.
As for the bad sectors causing a slow boot. It is most likely a contributing factor, but I am willing to bet that the hard drive is a 4200RPM model (most are) and as such will cause the system to boot slowly. So bad sectors + slow hard drive = even slower boot. You can pick up a 5400RPM at the least or even a 7200RPM one if you can justify the premium in price. If it is an older laptop, you may not want to dump too much money on it, so that is something you will have to factor in as well.