WALK!!! Paris is very...
WALK!!! Paris is very condusive to walking. The main landmarks the Eiffel Tower, the Arc de Triomphe, the Louvre, and Notre Dame. These are all definately worth seeing on their own, and once visited they serve as excellent landmarks for navigating the city by foot. There were so many places I loved to walk. There are two walks in particular that I would recommend.
First, any walk along the Seine is a delight. For this I liked to start at the Eiffel Tower and walk along the river until I reached Notre Dame. Feel free to stop at any of the bridges along the way and cross half-way to get some great vantage points of the city.
The next walk is a little busier, but still interesting. Start at the Louvre Pyramid and walk straight towards the Arc. You will pass the Tuilleries and the Place de la Concorde with it's Ancient Egyptian Obelisk before continuing straight up the Champs Elysées.
There are also many great areas to just explore blindly. These include the (1) Latin Quarter, specifically around Rue St. Michel, (2) the area surrounding Rue Moufftard (behind the Pantheon), and (3) the Marais which is an area on the Right Bank (North side) of the river and west of the Bastille. These areas are great to explore and will provide you with many pleasant surprises, as well as neat shops and restaurants at reasonable prices.
For people watching I would recommend any cafe with outdoor seating, and two parks, Parc Monceau and les Jardins de Luxembourg. If you just want to see the sights and you have time, do not miss the landmarks mentioned above, as well as the Musée d'Orsay, Sainte Chapelle (excellent stained glass from the 1200's...located on Ile de la Cité two blocks west of Notre Dame inside the Palais), and the Père Lachaise cemetary where many famous Parisians, including Jim Morrison, are buried. And, for those with a strong stomach, I would also recommend the catacombs (located at 1, place Denfert-Rochereau, just off the Denfert-Rochereau métro) and a tour of the original Paris Sewers (Les Egouts, located at the corner of Quai d'Orsay and Pont de l'Alma).
This list includes about half of what I would recommend but it is already overwhelming, so I will not add to it. Do not be afraid to walk without a plan. It's a great way to learn the city and see Parisian culture. Please email if you have any questions about details, etc. Bon chance! Getting a crepe from the same crepe stand everyday before class. By chance I discovered this wonderful crepe maker. He is actually a Greek man who works Paris during the warm seasons. I was never once disappointed in his crepes, and his prices beat everyone else too. When I last visited he was still in the same spot, so I recommend you pay him a visit if you want the best crepes Paris has to offer. You can get there by taking the #8 (purple) Métro to Ecole Militaire. From there, head away from the actual Ecole Militaire building on Avenue du Marechal Gallieni. Take your first left down Rue Cler, a pedestrian street/market area. Follow this for about half a block and on the left where this intersects Rue du Champ de Mars, there is cafe. I cannot recall the name of this cafe, but just in front of it you will find a crepe stand with jolly man in his early 60's with silvery-gray hair. I'd suggest Banana and Sugar or Lemon and Sugar, though any combination is delicious!