Here is a plan for visiting major sites of Paris chronologically.
1. Roman Lutetia to the high Middle Ages
This itinerary can be completed entirely on foot. Start at les Arènes de Lutèce (a Roman amphitheater off rue Monge). Walk from there to the Cluny Museum, whose basement is in the old Roman baths and whose main floors were once a medieval monastery. Explore the collection, noting the development from Roman sculpture through barbarian jewelry to Romanesque sculpture to the moving and sophisticated unicorn tapestries, which evoke the late medieval ideals of chivalry and gentility.
Walk north to the church of Saint-Séverin, noting the medieval street plan in that vicinity. Visit the church’s interior, focusing on the forest of Gothic columns in the apse. View Notre Dame across the Seine. Cross the bridge and visit the Conciergerie, the medieval royal palace. (You are allowed to see the exhibitions having to do with the Revolution of 1789, even though this is out of order.) Then enter the Sainte-Chapelle, whose walls of stained glass make it one of the finest displays of Gothic civilization in Europe. Finally, visit the heavily restored interior of Notre Dame and climb to the towers, bearing in mind that most of what you are seeing here (such as the famous gargoyles) dates only to the 1800s.
2. The Renaissance
If convenient, start at the church of Saint-Étienne-du-Mont, which was constructed continuously while the prevailing style shifted from Gothic to Renaissance. Even though the architectural vocabulary is all mixed up, the colors and scale are consistent, making the church interior a harmonious and lovely space.
Then ride the metro to the Marais and start in the Place des Vosges, a grand late-Renaissance planned space. Visit Victor Hugo’s house, just so you can get into one of the buildings of the Place. Exit through the Hotel Sully and consider visiting either the Musée Carnavalet or the Musée Cognac-Jay, each inhabiting a palace built while the Marais was at the height of its popularity, in the early 1600s. This is the Paris of the three musketeers. Walk toward the Louvre, whose eastern portion represents the late Renaissance. Visit the Renaissance painting and sculpture collections inside.
3. Louis XIV to Napoleon
This would be a good day to go out to Versailles. If you don’t want to make that trip, here is an itinerary for Paris proper: Start at the Invalides to get a flavor of grandeur, Louis XIV style. Walk along the Seine embankment to the Place de la Concorde, originally the Place Louis XV, whose architecture epitomizes the mid-1700s. (It then became the site of the guillotine during the Terror). Explore the Palais-Royale, which played a crucial role in the Revolution. Cross the river and walk to the Panthéon by way of the Sorbonne. Three classical domes, three Baroque or neoclassical interiors, and a lot of grand vistas.
4. The Industrial Revolution to Postmodernism
Start at the Musee d’Orsay and enjoy both the building (formerly a great train station) and the art collection. Make a detour to the Eiffel Tower. You could get a sense of the Paris of Boulevards and the haute bourgeoisie by taking a bus to the Parc Monceau and the Musée Jacquemart-André. Next stop is the Orangerie, which houses Monet’s Water Lillies from 1918 (a bridge from impressionism to abstraction). End at the postmodern Pompidou Center. Ride the external escalators to the top for the view, and look at the permanent collection of modernist art.