Seriously,
(1) Without pull or substantial chutzpa and guile you'll not get on the very "top" American club courses, unlike in UK when anyone can get on Muirfield, etc. on the open days for visitors.
(2) If you're not an experienced visitor to the new land, take some time to get to know it, for better or worse, apart from golf.
(3) The USA is a fairly large country, geographically, so I'd recommend not trying to move around much. Pick a spot and stay there. E.g., Pinehurst, although other than golf the North Carolina sandhills have only a little other stuff to offer the tourist (so combine with a trip to Myrtle Beach and its courses, to get the full flavor of the USA and its golf and culture).
(4) The American gun culture, a gift of our Constitution and the Supreme Court's interpretation thereof, is unique among so-called civilized countries and nothing to joke about. However, by knowing in advance where to venture and where not to venture, you ought to be safe, just as safe as you'd be from terrorists in other parts of the world.