It may be that other regional forms of English have stricter rules on this point, but in U.S. English it is not at all uncommon to speak of (for example) a "three-week trip." In fact, a search of the Google Books database—most of whose books and articles have been professionally edited—yields more than a hundred unique matches for that particular phrase. Here are a few examples.
From Roy Jenkins, Churchill: A Biography (Farrar, Straus and Giroux, 2001):
Then, almost as soon as he got back to Chartwell after this three-week trip [stopping in various parts of France], Churchill turned round and went back to Paris to deliver (in English) a much publicized lecture at the Théâtre des Ambassadeurs on the need to defend parliamentary democracy and the liberal state.
From Daniel Walker Howe, What Hath God Wrought: The Transformation of America, 1815-1848 (Oxford University Press, 2007):
A depressed and bitter president-elect managed to avoid the celebration that had been planned to welcome him to Washington at the end of his three-week trip from Nashville.
From Marc Leepson, Lafayette: Lessons in Leadership from the Idealist General (Palgrave Macmillan, 2011):
The Marquis de Lafayette's voyage back to France was not an easy one. Storms buffeted the Brandywine for a good part of the three-week trip. Lafayette, a veteran of rocky Atlantic crossings, arrived back home in good health and spirits.
From William Trimble, Admiral William A. Moffett: Architect of Naval Aviation (Naval Institute Press [Annapolis, Maryland], 2014):
Moffett initiated the selection process himself in September 1928 during a three-week trip that took him to the National Air Races in Los Angeles and on a tour of naval reserve air facilities at Long Beach, San Diego, San Francisco, and Seattle.
From these instances (which come from serious publishers that are by no means slack with regard to copy editing), it seems evident that any attempt to brand the phrase "three-week trip" inherently wrong must rely on a prescription about about what is proper, rather than a description of how often and freely fluent English speakers actually use the phrase and others like it.
As for the going and/or coming home question, a trip may be one-way or round, although most people take trips from a home base that they expect to return to.