All of this has been covered elsewhere, but most important is to have copies of the marriage certificate stamped and seen at TECO and to get your criminal record check sorted before you come and make sure that it will be valid at least through the time you plan to apply for your JFRV. Having your criminal record check stamped and authenticated doesn’t hurt either. Do a search on criminal record checks for Americans to get more detail on how to go about that. I seem to recall that it must be issued through a federal agency (the FBI), as municipal or local police checks are no longer accepted. The rest can be done here.
Once here, you can do the health check, get on your wife’s household registration, translate and notarize all supporting documents and submit the paperwork. The turnaround used to about a week to ten days, but this may have changed and someone else might offer more accurate info.
After all this is done, your wife will be able to list you as a dependent on her National Health Insurance and get coverage for you there. You do not need a work permit with residency through marriage.
Most of the infor is in the General Legal Matters thread under marriage: Getting married and applying for JFRV
Good luck!