How do I replace a fuse box to a circuit breaker panel?

The safest answer to your question, as asked, is to get a few quotes from electricians and pick the one you feel most comfortable with and have him do the job.

Note that service upgrades are a bit harder. As for some general guidance. There is pulling permit. Getting meter pulled to disconnect power. Having it inspected and meter put back into place. The meter will not be put back until it's inspected and passed.