Rue Rivoli dissects the Marais, the city's only district with many manors of the pre-revolutionary aristocracy. The area to the north of Rue Rivoli is much the livelier – where you can find both the ancient Jewish quarter and "gay Paris" – while the South is much more, well, Catholic. (Jim Morrisson died here in his flat on 17 Rue Beautreillis.)
For a walk, I recommend the neighbourhood of the Rue des Jardins St. Paul with its medieval courtyards that are filled with art galleries and restaurants. Ironically, the historic 4th also houses the city's most famous building of cutting-edge modernism, the eccentrically styled Centre Pompidou which hosts exhibitions of anything from modern art to cinema. The big plaza in front of the building is a popular location for musicians and other street performers – or even to have a picnic.