I lived in Fontainebleau for 6 months as a studenmt, which is an hour south of Paris so I got through there quite often. Obviously the Champs D'Elysees is worth having a look at, some nice restaurants up the side streets. The Left Bank is excellent too, some excellent restaurants. The Pompidou Centre and surrounding area are worth a visit.
The Louvre is ok if you're artistically inclined, but it's a lot to take in in one sitting. Have a walk over one of the bridges on the Seine, or along the banks.
If you really want to go to town and impress your young lady, wait until it's dark and take her to Place Trocadero, nearest metro (right next to it) is Trocadero - views of Paris by night DO NOT get any better than this; it overlooks the eiffel tower, PLace Concord (with all the fountains), etc, all of which are lit up at night and look spectacular. I took a young lady there to propose to her, and by jingo it worked

So I can safely say that Chicks Dig It.
I wouldn't bother with Pigalle / Moulin Rouge, as it's actually a seedy whoring district. Pere Lachaise cemetary, which Jim Morrison and Molliere (among others) are buried is worth avoiding.
Also, get a money belt - not being paranoid, but there are pickpockets on the Metro, and also working the crowds who stop to watch street theater, so mind and keep yer hand on yer ha'penny.