Paris is by far my favourite city, I even prefer it to London. I did an Art History degree and personally think that if you are just going to one gallery in Paris and are interested in art you should visit the Louvre, it is amazing and has a fantastic gallery with works by David, Gros, Delacroix, Ingres etc. The Orsay is also very good but it's mostly impressionist works although there are some beautiful paintings by Degas there and Toulouse Lautrec, there is also a wicked glass floor, there is also a beautiful painting by Cabanel on the first floor.
I would also go to Sacre Coeur, there in the square behind is a lovely inexpensive but lovely restaurant called Chez Eugene, the atmosphere is fantastic and the food is surprisingly good, the decor is also gorgeous.
The catacombs are worth a look but a little creepy as the tunnel goes on for ages and you feel as though you are the only person there! The Luxembourg gardens are also lovely, The Picasso museum is worth a look and the Pompidou is great if just for the view over Paris from the top of the escalators, on a clear day you can see the Eiffel tower and Sacre Coeur.
Notre Dame is also a must, it is breathtakingly beautiful, a trip to Versailles is also worth doing, it takes about 40 mins on the train. There is also a street near the University, (can't remember what it's called) but it has a lovely house designed by Le Courbusier.
You should also make a trip to the Opera House and see Chagall's ceiling as it is so so beautiful. The observatory is also good and the revolutionary museum where you can see Napoleon's costumes and things.
You will seriously love Paris, if there is anything else you would like to know please ask!