The best thing to see in Paris are, obviously, the art-viewing possibilities, as many as such a developed culture could offer - the museums and galleries are crowded with amazing pieces. Then there is the food, as amazing and multicultural as could be, and the romance is in the air.
Quick Tips:
The Musee Galliera, The Musee D'art Moderne de la Ville, and The Palais of Tokio, such as the Cartier Foundation, are three amazing, not-so-known places that have a great museography and a cut of the edge in contemporary art. There are really amazing, mysterious, and luxurious tea houses on the 4eme. and 6eme., and just walking all through Champs Elysee might present a new perspective of life and evolution.
Best Way To Get Around:
If I were to recommend a way of aknowledging this city for the first time, or just to really get to know it, it has to be either walking or by bus. The city is not so big, so you an go across it in a whole day of walking, or just by regular buses, which are comfortable and not so expensive if you buy a weekly pass. You can see the city as the French do.
I don't think the subway is the best, since most of the routes are below the city - you´ll get around fast, but will only see tunnels.