Here are some ideas for seeing the city without much cost:
Parc Buttes-Chaumont, a man-made park with hills and dales
Père La Chaise Cemetery and Montparnasse Cemetery (metro same names) have the graves of many famous people, including Jim Morrison and Oscar Wilde in the former and Sartre and Beauvoir in the latter. See the book, Permanent Parisians (cheap second-hand copies available on Amazon) in the Condé ESA library for great stories about some of the famous and not-so-famous grave sites.
Sacré-Coeur and the Place du Tertre in the heart of Montmartre (Metro: Abbesses): this is the touristy center of the area, like what you saw in Amélie. As it's a tourist area, be careful about pickpockets!
Parc Monceau, a manicured park in a rich neighborhood and near the Asian art museum, Musée Cernuschi and the Musée Nissim-Camondo (paid entry, beautiful grand home decorated with 18th century furniture)
The Canal St Martin near the Bastille. This company has a great boat tour!
The food shops on the rue Mouffetard, near the Panthéon (looking and smelling are free!)
The Promenade plantée, an elevated train line remade into a park. Start at Avenue Daumesnil near the Bastille.
Music concerts by the bandstand in the Luxembourg many afternoons
City museums are free (national museums like the Louvre or the Musée d'Orsay are not). Here are some free museum examples:
The Musée Carnavalet, history of Paris. In the Marais, métro St Paul
Musée d’Art moderne, métro Alma-Marceau or Iéna
Musée Cernuschi (Asian art) near the Parc Monceau
Musée Cognacq-Jay with 18th century art in a lovely home, metro St Paul
Musée Zadkine, 19th & 20th century near the Luxembourg
Petit Palais, near les Invalides, 19th & 20 century European art
Musée Curie, the labs of the scientist
Here is the complete list of city museums:
http://parismusees.paris.fr/en/les-musees-de-la-ville-de-paris
Take in a movie (check first to see if the theater has "la climatisation" ("clim")
Go (window-)shopping in one of the large department stores by the Opera (i.e. Le Printemps, Les Galeries Lafayette)
Read a book in the Luxembourg Gardens-- find a shady spot near the grass
Visit the Louvre (some areas have A/C) or Musée d’Orsay (A/C throughout)
Visit a church-- those medieval architects knew what they were doing! The old churches are very cool inside.
Go swimming! There are many municipal swimming pools or you can visit Aquaboulevard (an indoor water park)
Use your Navigo to go out to the countryside near Paris! If you are lucky, your train/RER will be air conditioned! There are lots of walks and hikes just outside of the city.