Barbès – Rochechouart (Paris Métro)
Basilique du Sacré-Cœur
Bibliothèque Nationale de France
Belleville
Canal Saint-Martin
Cité des Sciences et de l'Industrie
Crazy Horse (cabaret)
Île de la Cité
Jaurès (Paris Métro)
Maison européenne de la photographie
Musée du Louvre
Parc des Buttes Chaumont
Palace of Versailles
Palais de Justice
Palais de Tokyo
Porte de Clignancourt
Seine River

Belleville is great. North of Paris in general is great for off the beaten path photography. If you want to take photos of the hustle and bustle of Paris, stop at Barbès-Rochechouart on line 2 or 4. I don't know when, but I think during weekends, they have some kind of crazy market by the Metro station; people selling fake packs of Marlboro smokes and weird fruits and vegetables. It's like you're in Timbuktu or something.
I thought Sacred Heart was great. I loved the view from there. It's a long confined stairwell to the top, and it smelled like piss, but what a view of the city. Nice shots of the tower, fantastic panoramas if you're into that kind of thing; totally worth the trek to the top, even if slightly claustrophobic.
For actually taking photos, I really liked the site of the Bibliothèque Nationale de France last time I was there.
Take the Metro to Jaurès on line 2. Walk south by the Canal St. Martin. This is probably my favorite place in Paris. Locals hang out by the canal. There are some really great cafes, and the area is perfect for street photography. And you can walk all the way down to Le Marais.
One place to recommend for science-type photos is of course Cité des Sciences in the northwest east.
Go see a show at the Crazy Horse! It's expensive but you get what you are paying for. You can forget about pulling up your camera in there, but the show is beautiful and classy.
I really, really loved the area of Île de la Cité. That little island on the Seine is just amazing. Add to that the surrounding buildings of the police and stuff. My favorite being the Palais de Justice. At night it will take your breath away. Plus there was a bar right near there that was fantastic; friendly bartender/owner who was native English speaker and a nice local crowd.
If the weather is too bad to be outside and you want to do something photography related, be sure to visit the Maison européenne de la photographie. Depending on what expositions they have on, you can easily spend an entire morning or so in there.
Montmartre - excellent views of the city. Grab some grub and picnic on the lawn.
Louvre - incredible art collection. I've never been particularly impressed with the Mona Lisa. I was blown away when I saw it in person.
Another spot I like is the Louvre Museum after it closes or very early in the morning. The glass pyramid is very cool; If you are lucky to have good light you can get really interesting long exposure shots there.
Parc des Buttes Chaumont. It's really, really wonderful.
Versailles - a bit outside Paris, but absolutely stunning. Felt unreal.
Besides the obvious in-city things, go to Versailles and go to Montemart near the Sacred Heart Cathedral; great little area for some street photography. Explore and keep your hand on your camera.
Palais de Tokyo for example is a nice museum.
Take the Metro to Porte de Clignancourt on line 4. Walk north to the flea market, Puces St. Ouen. There are some pretty amazing things here. You can buy everything from taxidermy monkeys in cages to actual pilot seats. And it's all gloriously French and begging to be photographed.
Definitely Clignancourt and St. Martin. If you want color.

If you're in the area, head behind the Sacre Coeur and find a restaurant called Sale e Pepe. Wonderfully authentic Italian food. You get a set menu, and it's all pretty casual, the server comes out with the pan and just scoops pasta on your plate, and comes back when you're done to see if you want some more.
You should get a bike. There is no vehicle better suited to this comparatively small city than a bike if you stay a bit longer. In the weekend you can take bikes along on the RER to explore the nicer outskirt (ranging from Conflans to Versailles.
Buy an anti-theft camera strap. It's a normal looking strap with a braided steel middle; it's invaluable. One of my friends had her camera stolen in Lyon in 2003 by a thief with a razor blade and a scooter in the middle of the biggest tourist area at 2 in the afternoon. One plus side is that you no longer have the obnoxious CANON/NIKON logo screaming at the world.
Sun-sniper has a very nice strap (similar to the r-strap) with a wired center.
I love the PacSafe CarrySafe 100. LowePro also does a nice one that I've been using for some time, but I can't recall the name.
Theft: be careful. We were in the Metro station and my friend felt her backpack getting tugged on. Her husband tried to grab the kid who had his hand in her backpack, but the kid just ran off. Tourists are easy marks in a city like Paris.
Don't fall into the hipster trap. "I'm too cool for school to do the tourist thing.” If you've never been to the Eiffel Tower, then GO to the Eiffel Tower, damnit. Once you do the tourist thing, you can then get off the beaten path.
I would check out the cemeteries. If you stay away from the largest one, you can do a fair job avoiding the tourists. Also, a few of them offer commanding views of the city.
Also, if you have a fast lens, I would recommend checking out the Metro tunnels on the weekends. There are often great indie/blues musicians there that provide a great subject and the lighting can sometimes be interesting. Watch out though, the crowd is moving fast so you probably can't bring your tripod.
Start to learn basic French phrases immediately and how to interact with cash. They hate when you expect them to speak English. This leads to shitty service. Most people there are great but, watch out for gypsy beggars (kids). They are not begging; they want your passport and money and will stab you in the ass. If a guy walks up to you in a restaurant and starts to read poetry to you in French he will expect to be paid - seriously. Do not let anyone talk to you pretending to want to practice their English or being overly friendly. Be smart and have a very secure bag.
The Metro is an awesome place to shoot; really beautiful down there. You could spend days there.
Photography laws for France are pretty liberal as in you can take picture from public spots, however not of policemen while in duty or military installations.
Watch out for the dust/dirt . I spent a month in Paris last year shooting 10 hours a day all over the city and came back with a sensor so dusty that I wanted to cry. Paris is awash in very fine dust because they use it in many of their parks and courtyards. No matter how careful you are changing lenses, odds are you'll end up with some nasty spots. I use an arctic butterfly brush, but after Paris I had to breakdown and buy a sensor stamp to get things back in order.
In Gare De Lyon I set up my tripod and started taking pictures, then two cops showed up and asked me if I was taking pictures. I politely said no and folded the tripod and put it in my bag. Then they went away. It was very funny.
If you like walking, just keep walking in the city and you will be exposed to many different scenarios. As far as I could see, most of the city looked safe.
In regards to photos: THE WHOLE CITY! Seriously, the place is a gorgeous goldmine of photography. There is such a mix of old-world history and modern culture, the dichotomy is astounding.
Just grab a map and explore. The subway system is super easy to learn, and takes you anywhere you would want to go. I found just exploring the city to be a blast and pretty safe except for really late at night.
Wear comfortable shoes (Clarks). You'll be walking A LOT! I lost 10 lbs. my first week from all the walking
Don't eat at tourist traps. There are a lot of crappy restaurants in crowded areas. Grab a crepe when you're in a rush, and eat at hole-in-the-wall joints off the beaten path. You will be eating for at least an hour. There is no such thing as a fast meal in Paris.
Ignore the gypsies. Seriously, just keep walking. Seriously.