There are several long trails that cross most, or all, of the USA from north to south: Appalachian Trail, Pacific Crest Trail, Continental Divide Trail. These are listed in more or less in order of difficulty. Do some googling for information (each of the trails has its own association) and read some journals on Trailjournals.com or Postholer.com. Our own BrianLe (Gadget) has completed all three of those north-south trails, and his journals (on Postholer) would be good to start with.

As for medication, you'd want to have refills arrive at each supply stop. If you have problems forgetting the medication, or have episodes of illness even when regularly taking the medication, it would be a good idea to have someone hike with you. Others with medical difficulties, some severe, have completed these trails.

To start, what you really need is to acquire backpacking experience and skills with a number of short trips before you start on the long trip.