I will surely not pretend to write a "Best and Worst of Paris" post. Don't know enough, wasn't there long enough, and many others have done it better than I could even if I knew more and had been there longer.
But, I can give some brief info on the things that were best in a really short visit. If you are having to make some choices, reflecting the need for time trade-offs, here are some suggestions.
Do / Don't pairs
1. DO: Arc de Triomphe / Champs Elysees. Climb the stairs on the Arc, and get a great view of Paris, including the Tour Eiffel. Then walk along the Champs Elysees, down toward the Place de la Concorde. Careful, though, no restaurants anywhere near the Place de la Concorde. Eat before, on the C.E., or later.
DON'T: Wait in line and climb the Eiffel Tower. Sure, it's pretty. But the view from the Arc is perhaps even better, because (1) the streets radiate out from that point, (2) you can SEE the Eiffel Tower from the Arc, and (3) the line for the Arc, is usually nothing, while the wait for the Tower can be 3 hours. If you have only a short trip, no way it is worth it to wait with all the other American and Canadian fanny-pack-and-Birkenstocks E-Tower crowd. Just say no.
2. DO: Go to the Picasso Museum for three hours (it opens at 9:30 am and is CLOSED on Tuesdays), have lunch, and then go to the Musee d'Orsay (CLOSED on MONDAYS, open late on Thursdays). If you visit each of those two museums, and spend three or fours hours each, you can actually see some really wonderful things. I happen to be very interested in Picasso, but even people who aren't (such as my good friend Duke Philosophy prof Alex Rosenberg) say that you should see the Picasso Museum to see the outline of change in art and society in the 20th century. A fine little museum. And the Musee d'Orsay is just outstanding. These two are a decent walk apart, or a pretty quick Metro ride.
DON'T: Go to the Louvre. Sure, you can go see "la Mademoiselle," and wander around the exhibits for a week, if you are a student of art history. But the Louvre is not really a museum, or even a collection of museums. The Louvre is a library of the history of visual arts, with a strong bias toward French works. That makes it invaluable for scholars, artists, and the truly knowledgeable. But it is overwhelming to the rest of us. If you only have a short time, give the Louvre a miss, unles there is a particular exhibit that interests you.
3. DO: Use the Metro. Just do it. It's not hard, and it is a big help. The "Information" guys at the stations are actually helpful. Here is a web site that has lots of information that I found useful.
DON'T: Take taxis, or walk too much. Either way is stressful. We took one taxi trip that was straight out of a movie. We wanted to get to a museum that was going to close soon (don't ask; it was my fault), and took a taxi at 5:05 pm on a weekday, in the medieval rabbit warrens of the 3rd Arrondissement. Now, what could POSSIBLY go wrong with that brilliant plan? Taxi driver did his best, fighting his way out to a main road, cutting across four lanes of traffic, pulling out into opposing traffic. He gave up, crossed the Seine, drove on the curb and the bus lane. Then he recrossed the Seine. Then we got stuck in REAL traffic and he told us to get out and run the rest of the way; it's hopeless. We made it to the museum, but... We also otherwise walked and walked and walked...and got too tired and grouchy. Plan your walks, with a map, along streets that are interesting, and take the Metro otherwise. It's Paris, not Bataan. And, if you do walk, carry bottles of water that you buy at groceries. You can get bottled water for 1 Euro at groceries, but drinks cost 3 Euros and up at cafes. Hard to stop "just for some water" without feeling ripped off.
4. DO: Take a boat ride on the Seine. Do it on the first day, in the afternoon. You get a better feel for geography, and history. And you get to sit down for a while, if you are tired of walking. Just admit you are a tourist, and take the Bateaux-Mouches route (an easy walk from the Champs Elysees or the Place de la Concorde). There's nothing sophisticated about it, but the boats leave every 20 minutes and you don't need to have a plan or reservations.
DON'T: Go to science museums, zoos, aquariums. They aren't very good, by American standards. Everything is written in French (that's fine, by the way, since it is IN FRANCE, but that doesn't mean non-French speakers will get anything out of it), and this is not the comparative advantage of Paris, anyway.
Finally, a lagniappe: The easiest and best triumvirate of things to see are on the Ile de la Cité. You've got the Conciergerie, the Saint-Chapelle, and Notre Dame cathedral. None of these is more than a five iron apart, even if you hit it fat. And a nice combo of interesting, beautiful, and informative. And plenty of places to eat or have a drink just across the river to the north, in the 4th Ar. part of the Marais.
More thoughts on a short trip to Paris: Paris in two days...