The Patisserie Viennoise at 8 rue Ecole de Medecine near Odeon (in the 6th) is reputed to have the best hot chocolate in all of Paris. Chocolate maven, David Lebovitz swears by it. This is a tiny pastry cafe (in operation since 1928) on a slim sliver of a street. Easy to walk by, but try not to miss it.
Rodin Museum, on the rue de Varenne in the 7th arrondissement, just to the side/behind the Invalides Museum. It was once a private residence, but then became both studio and residence (Isadora Duncan and Jean Cocteau once lived in this building) for the sculptor Auguste Rodin. The museum is now devoted to Rodin's work and houses what must surely be the world's largest collection of his oeuvre. Rilke worked as his secretary for a number of years and had an office in this building. The gardens behind the house are a very peaceful place to stroll.
Luxembourg Gardens, what can I say. 66 acres in the middle of Paris. Pure heaven.
The Palais Royal, in the 1st arrondissement, just north of the Louvre. Its interior courtyard garden, with its perfect double rows of trees has got to be the most gorgeous and peaceful oasis in the heart of Paris. In the late spring or in the summer, try making a reservation at the Resaturant du Palais Royal and ask to sit outside to enjoy the gardens during your meal. You can also walk all the way around the interior arcade of stores and window shop.
Take the métro to the Passy stop from the left bank to the right bank and as it crosses the Seine, the train actually emerges and runs outside for a stretch over Bir Hakim bridge. If you look out the right-hand window, it's a lovely view of the Eiffel Tower -- the Passy stop is where you would get off to walk to the Musée Balzac, Balzac's little Parisian house, now a museum.
The Passage Vero-Dodat is located between the rue Jean-Jacques Rousseau and the rue Croix des Petits-Champs in the 1st arrondissement. It is, like the Galerie Vivienne, sort of behind the Palais Royal and a very good example of the 19th century covered walkways of Paris. Unlike the Galerie Vivienne, the Passage Vero-Dodat has a rough-around-the-edges look that feels much more authentically 19th century. The Galerie Vivienne is a horse-shoe shaped covered walkway that you can access just north of the Palais Royal on the rue des Petits-Champs. It is the most beautifully restored ‘passage’ -- a precursor to our malls. This covered walkway houses the gift store for the Biblioteque Nationale, the Grand Colbert (a good restaurant), and many interesting stores. There is A Priori Thé, too, the tea salon which is a perfect resting spot after a long walk: wicker chairs, wooden tables, soft lighting, and delicious teas and pastries as well as a lunch menu.
A small, out of the way church that I like in Paris is (also behind the Palais Royal) the Basilique Notre Dame des Victoires on Place des Petits Pères in the 1st arrondissement. The walls are covered with plaques thanking St. Therèse for answered prayers. It is a beautiful, small church -- quiet neighborhood. (There's also a really good bakery right across the street!)
In front of the �glise St. Sulpice, in the 6th arrondissement, is a large, well-proportioned square with a giant fountain in the middle. It is a most pleasant place to be. There is only one small café (Café de la Mairie) on this square but many of my friends love the simple, truly Parisian atmosphere of the place. You can sit indoors or at one of the tables that spills out onto the sidewalk in the spring and summer. It is a simple, easy place to hang out, and not overpriced like the very trendy Café Deux Magots or Café de Flore, nearby.
In the spring or summer, since the sun tends to set very late (10pm), a post-dinner walk is nice and one that takes you to Notre Dame Cathedral and then across the Pont St. Louis onto the Ile St. Louis. Once there (it's a very small island), head for Berthillon Glacier at 31, rue Saint Louis en l'Ile to buy a cone of their amazing ice cream. Other cafés and restaurants on the island sell their ice cream to passers by. They have about two dozen flavors.