The number one thing to remember about Paris is that the French are proud of their language, and they get sick of so many stupid Americans coming through and inundating their capital. It's a good idea in any city you visit to pick up some basic phrases, but in none moreso than Paris!
Metro: The Metro is really amazing, and goes everywhere. There are two levels of it: the local Paris metro, and the deeper level, which contains the commuter trains. Buy a ten-ride packet for travel within the city, and remember that their tickets are so tiny that they must have been designed to be easily lost.
Places to stay:
hostel Quartier Latin
Decently cheap, and is located a few blocks away from Notre Dame. There are a couple bakeries right around the corner, and a fruit market that sets up very near it, so you can eat cheaply. Bring a towel (but any self-respecting traveller will have brought one anyway). They're pretty clean.
Art Galleries:
Rodin Museum
Beautiful Rodin sculptures pepper the finely-manicured gardens. This is a fantastic place in which to bring a camera.
Picasso Museum
This place is horribly hard to find, making the experience of wandering the gallery even more pleasant. I am not Picasso's biggest fan, but the architecture of the building alone made the trip worthwhile.
the Louvre
A big "duh" of Paris. Get there early to miss the crowds, and avoid the Italian wing. We found the entire museum deserted, except for the Italian wing and especially the three hundred square feet around the Mona Lisa. There is so much to see here that you may wish to devote more than a day to it.
Centre Pompidour
Paris' fantastic modern art museum. On display while I was visiting: a huge, disembodied nose which spun in circles, hummed, and smelled pleasantly of spices; also a video of naked women painting themselves blue and throwing themselves against a blank canvas, courtesy of Yves Klein. This is a great place to go if you've just come from Spain and Italy and your eyeballs are bleeding from seeing so many "Madonna with Child" paintings.
Restaurants:
One puzzling aspect of my trip is that, despite Paris' fame for its food, I encountered few good restaurants. I do have one very good recommendation, though.
Cafe Beaubourg
They serve the BEST hot chocolate I've ever tasted. We would eat there every day for breakfast, and just make that our big meal of the day, as it was the epitome of perfection: soft and creamy eggs, delicately-toasted bread, concentrated lemonade to offset the sweetness of the hot chocolate, and a croissant whose butter was still melting from the oven.
Chartres
Chartres is a city about an hour's train ride away from Paris. Notre Dame is gorgeous, but far too crowded. Chartres is bigger than Notre Dame and entirely deserted, AND you can go into the towers for free!