The itinerary is actually six itineraries intermingled with each other. Each itinerary is for the subway system of a different world city. As hinted in the instructions, at each stop, you must change trains (and only change trains at listed stops). The routes given use all the trains of each subway system except one; the first letters of those trains (read in the order each city first appears in the scrambled list) give the final answer, UPTOWN. The train listed in parentheses after each station below is the train you take to get to the next station in the list.
Paris: U (Un [1])
Franklin D. Roosevelt (9)
Oberkampf (5)
Stalingrad (7)
Jussieu (10)
Duroc (13)
Invalides (8)
Madeleine (14)
Bercy (6)
Raspail (4)
Montparnasse Bienvenüe (12)
Saint-Lazare (3)
Villiers (2)
Belleville (11)
Goncourt
London: P (Piccadilly)
Great Portland Street (Circle)
Embankment (Bakerloo)
Oxford Circus (Central)
Mile End (Hammersmith & City)
Farringdon (Metropolitan)
Finchley Road (Jubilee)
Canada Water (East London)
Shadwell (Docklands Light Railway)
Bank (Waterloo & City)
Waterloo (Northern)
Warren Street (Victoria)
Victoria (District)
Bow Road
Moscow: T (Tagansko-Krasnopresnenskaya)
Kitai-Gorod (Kaluzhsko-Rizhskaya)
Prospekt Mira (Koltsevaya)
Park Kultury (Sokolnicheskaya)
Okhotny Ryad (Arbatsko-Pokrovskaya)
Kievskaya (Filevskaya)
Aleksandrovsky Sad (Serpukhovsko-Timiryazevskaya)
Sevastopolskaya (Kakhovskaya)
Kashirskaya (Zamoskvorestkaya)
Novokuznetskaya (Kalininskaya)
Ploshchad Ilyicha (Lyublinskaya)
Pechatniki
Washington, DC: O (Orange)
L'Enfant Plaza (yellow)
King Street (blue)
Metro Center (red)
Fort Totten (green)
Navy Yard
New York: W
Rockaway Park (Rockaway Park Shuttle)
Broad Channel (A)
Broadway Junction (L)
Union Square (R)
Jackson Heights-Roosevelt Avenue (7)
Grand Central (42nd Street Shuttle)
Times Square (N)
New Utrecht Avenue (M)
Hewes Street (J)
Crescent Street (Z)
Chambers Street (5)
East 180th Street (2)
14th Street (F)
Bergen Street (G)
Hoyt-Schermerhorn (C)
Cathedral Parkway (B)
Tremont Avenue (D)
Columbus Circle (1)
Van Cortlandt Park (9)
Penn Station (3)
Crown Heights-Utica Avenue (4)
Franklin Avenue (Franklin Avenue Shuttle)
Prospect Park (Q)
Canal Street (6)
51st Street (E)
Queens Plaza (V)
Broadway-Lafayette (Grand Street Shuttle)
Grand Street
Tokyo: N (Namboku)
Hasune (Mita)
Jimbocho (Shinjuku)
Morishita (Oedo)
Hongo-sanchome (Marunouchi)
Ikebukuro (Yurakucho new line)
Kotake-mukaihara (Yurakucho)
Nagatacho (Hanzomon)
Omote-sando (Ginza)
Shimbashi (Asakusa)
Nihombashi (Tozai)
Kayabacho (Hibiya)
Kasumigaseki (Chiyoda)
Machiya