One could easily spend a two weeks in each of Paris, Rome, and London, and still have lots of things to do or see. Most major museums require reservations - most can be obtained online.
Rome: A lot of art can be seen for free at various churches. See Caravaggio paintings in the church Santa Maria del Popolo, the north side of Piazza del Popolo. Go to the Museo Borghese (at the Villa Borghese) to see the incredible Bernini sculptures, Apollo and Daphne, and The Rape of Proserpina. You must have reservations, which can be obtained online.
Paris - See the stained glass windows in St. Chapelle in the Palais de Justice on the l'île de la Cité. These incredible windows survived the revolution intact. See them on a late afternoon, or in conjunction with a an evening concert held in St. Chapelle. Sample some of the food sold in the pedestrian shopping street Rue Montorgueil - just north of Les Halles. If you are going to the Musée d'Orsay (you need a reservation if you don't want to stand in line forever) buy your ticket/reservation at a FNAC store. Skip the Louvre, and visit the Musée Marmottan (Monet) instead. If you like Monet, visit the Musée de l'Orangerie - has his immense water-lily murals.
London: As soon as you arrive, get a copy of Timeout magazine - lists everything that is going on in London. Early in the day, go to the official half-price theatre booth (TKTS) at the south side of Leicester Square, and get cheap tickets to shows. Walk through Hampstead (on the Northern Line) for 18th and early 19th century buildings. Visit the Borough market.