On Ile de la Cité, there is the famous Cathédrale de Notre Dame de Paris. This is the heart of Paris and one of the best examples of Gothic architecture. Inside there are the Rose windows and the trésor, which houses works of art and liturgical objects. Amazing views of this Cathedral can be seen from Square Jean XXII.

Musée d'Orsay houses the national collection of sculptures, paintings and works from the 1850's to 1914, which includes impressionist (Monet, Renoir, Degas), postimpressionist (Gauguin, Van Gogh, Matisse) and some art nouveau pieces.

Musée Auguste Rodin, set in a 18th century residence, and the gardens around it, are considered by some as one of the most relaxing places in Paris. It contains bronze and marble sculptures by Rodin (works such as The Kiss and The Thinker) and Camille Claudel.

The Château de Vincennes, to the north of Bois de Vincennes (the Vincennes Wood), is a place to see if only for the moat, dungeon and fortifications surrounding it! Louis XIV spent his honey moon in the Pavillon du Roi, which flanks the the royal courtyard. There are various guided tours which will take you inside the Chapelle Royale or you can take a leisurely walk around the grounds by yourself.

Useful Links:

Musée de Louvre

Centre Pompidou

National Monuments
(i.e. Cathédrale de Notre Dame de Paris, Château de Vincennes, Arc de Triomphe)

Musèe Orsay

Musée Rodin