I would stick to a European trip if you want to absorb culture. Get a euro train pass via Deutsche Bahn. Tons of places to hike regardless the country. If I were you, I would fly into Munich or Prague and work my way to the west coast. Anything further East can get sketchy. I would make my way towards Belgium too enjoy a true evolution of beer. Then head south into France for a wine experience. Then head down towards Spain/Portugal to hit up some beaches and finish in Rome to get a last bukkake of culture.

I would recommend area focused visits of the US otherwise it gets too expensive or way to long to travel.