Naturally, the Grants were no ordinary tourists, nor even the American equivalent of grand tour travelers. They met royalty and other heads of state, attended functions in their honor, and attracted crowds of people at parades and other events. Newspapers back home covered their movements in detail, and the public read with fascination.
Grant's love of travel ran deep and, according to his Personal Memoirs was in fact one of the reasons he went to West Point. "Besides the argument used by my father in favor of my going to West Point -- that 'he thought I would go'--there was another very strong inducement. I had always a great desire to travel. I was already the best-traveled boy in Georgetown [Ohio]..."
A short description of the Grant's world tour is here, at the American Experience web site.