Rue Pot de Fer (5th arrondissement), near the Monge Place metro stop. Pedestrian only street of tiny hole in the wall restaurants.
Marais-good cafes. Check out the BHV across from the Hotel De Ville. Decent shopping in the BHV.
Galleries Lafayette-behind the Opera House. Beautiful shopping center. Worth a stroll into just for the ceiling alone.
Saint Chappelle: located inside the Palais de Justice, pretty close to Notre Dame.
On Sundays, the flower market near Notre Dame is cool to see.
Montmartre at night time is fun. A bit dicey but nice view, cool street scene. Sacre Coeur is a neat basilica.
Hit up any random patisserie, get a pain au chocolat and enjoy the smells.
Tower view inside Notre Dame. Highly recced by my wife's friend.
Any outdoor markets.
Grab a bottle of wine, some bread, hang out near the Eiffel Tower on the Champs de Mars and watch people.
Check out Napoleons giant tomb in the Les Invalides.
Hell, I had fun just going in their grocery stores.