@jcblubaugh, I am not a heart transplant recipient, but I am a liver and kidney transppant recipient. I received my transplant at Mayo Rochester which is 1750 miles from home. Perhaps I can shed a little light by my experience.

When in the hospital, the nurses will teach you everything that you will need to know before release to outpatient care. Then outpatient care will continue with labs, tests, and doctor visits. It is a busy time. By the time that you are released from outptient care to go home, you will have learned how to manage your care. One of the first things that we (me and husband) were told was to make an appointment with my local doctor, and be sure that he has the medical information from Mayo. I remember being really uncomfortable to leave the safety of my Mayo team, but upon my departure my nurse coordinator said that She was only a phone call away! And that is the truth!!
Once home, and settled in, I was able to relax a bit and even became emotionl about all that I had been through. As a heart patient, I expect that you will be able to learn more from a heart patient. For me, I had frequent labs every week, every 2 weeks, currently 3 months, and as medically necessary. My labs were (and still are) drawn locally and sent to Mayo. After Mayo read them, they will let you know if any changes needed. I returned to Mayo at 4 months, and then annually. My primary care doctor takes care of dirrecting my health needs locally, but the transplant care is totally managed by the transplant team. My primary care doctor, and any specialist, will contact Mayo when necessary (or I will contact Mayo myself!)
One extremely helpful thing that I would suggest, is for you to use the Mayo Specialty Pharmecy if Mayo Jacksonville offers it. Meds come right to your door, and refills or changes are efficiently provided by Mayo doctors and pharmacy.
I hope this helps.