I did a flight in an Archer from LA to Key West and back again.....was a while ago, and I did it in around 3 weeks....had a fantastic time!
Great trip, top tips: buy a Rand McNally road atlas, circle where you wanna see. Buy the AOPA "telephone book" which lists all the FBOs and what they offer. Get hold of all your (aviation) maps early on. Plenty of free nights in great accommodation and free cars available if you pick the right airports. Also look at the height of them there mountains, some are very large and you might want to plan your route carefully. Plan your flights. Every flight. Don't cut corners. Be safe. Have options and alternatives and back up plans and back ups to your back up plans. Learn how to file a flight plan and do it every flight. Flight plan your route and basic info the night before.
How much time have you got, how much flying have you done so far, have you done any in the USA, have you flown into any big airports? What licences will you have when you do it? (I had FAA PPL and single engine IR - the IR bit (and the night bit) helped loads). Are you planning on trying to do a bit of sightseeing as well? What sort of things are you into? I went to loads of great places, things like New Orleans for the night, Kentucky when the Derby was on, Key West, Roswell (!), etc. etc. Been to big airports, small airports, high airports.
Anyway, if you want more info feel free to PM me.
PS I have done a fair bit of flying in the USA, been over a few times for hour building before getting airline job and been back regularly since. I can dig out my logbook if you want some good routes/ times and so on.