Definitely get a city guidebook before you go, and plan out what things you plan to see in advance. Time flies, and you don't want to miss out on something important!
Buy a day pass for the Métro (subway) -- it's the easiest way to get around the city, and it's easy to navigate, even if you don't know French. Do NOT attempt to drive in Paris. Repeat: DO NOT ATTEMPT TO DRIVE IN PARIS. It's a maze of one-way streets, intersections with multiple exits NOT at right angles, and CRAZY drivers. Drivers in Paris make Washington DC drivers look like DMV examiners.
You can survive without speaking French -- you'd be surprised how well you can get by with pointing and gesturing -- but you should know some basic phrases so you can read stuff on your own: directions, prices, menus, warning signs. Many people speak English, and despite what you may have heard, there isn't an automatic attitude towards Americans. However, I would strongly advise you to behave in a low-key manner, dress like the natives (no American-branded sneakers, no baseball hats or sports clothing, no white socks), and basically try to blend into the crowd. You never know who MIGHT have a problem with you -- so why stand out any more than you have to?
Paris pickpockets are legendary. Keep your valuables in your front pocket, and check it at least every minute or two. Do NOT allow yourself to be touched -- if you get bumped, never assume it's an accident, and check your pockets instantly. Consider buying a cheap "decoy" wallet to keep in your back pocket. Do not wear flashy or expensive jewelry. Hold on to cameras and purses very tightly.
As for what to see -- the typical touristy stuff can be really nice. The Eiffel Tower does provide a nice view of the city, and it's in a beautiful central mall which is pretty to walk through. However, it's also crowded as hell. I went to the Louvre, and although pretty, I should have known better given my taste in art -- I MUCH prefer more modern art. If you only get to one art museum, it's probably a toss-up between the Louvre and Musée d'Orsay, which houses works by many of the familiar Impressionist and Post-Impressionist painters. I had a chance to go to the Catacombs, and they were a sobering and impressive experience. (The entrance is hard to find since it's so nondescript -- you need to exit the Métro at Denfert-Rochereau and look around for a little bit.) I didn't see much of Nôtre-Dame since it was Sunday morning and services were going on, so it was inappropriate to enter. (That didn't stop a shitload of tourists from going in and taking pictures DURING the service, though. I wanted lightning to strike them down BADLY, but God must have been busy that day.)
Stuff I didn't get to visit, but wanted to badly: Musée d'Orsay, Montmartre (artist's colony at the top of a hill; right next to the famous cathedral Sacre Coeur), the Picasso Museum, Centre Pompidou, and Versailles (which isn't in Paris proper, but is not far from the city).
Finally, the food is SUPERB. Enjoy a coffee (mmmm, European coffee) and a pastry, and people-watch for a little while at a cafe. I won't even bother to make dinner recommendations, but I DO suggest having wine with every meal. French chocolate is also very good and underrated.
Hope you have an awesome time!