In the south end of the city, you cn walk through the catacombs - tunnels in the limestone where they piled the bones removed from the old graveyards after the revolution. Bones, skulls, and more.
Go everywhere by subway (metro). Unless you're from a place like new York, the subway itself is a different and interesting maze to follow.
Climb to the top of the Arc de triomphe. Watch the crazy traffic in the circle around it.
I assume you're seeing the usual sights - The Louvre, including the Mona Lisa, Venus de Milo, Nike - Winged victory, etc. The Eiffel tower - visit the top. Notre Dame - you can climb the towers (no pouring hot oil on the tourists!); You can rent a rowboat in the Bois de Vincennes or Bois be Bologne - see the medieval castle in the Chateau de Vincennes - should be open by now.
You can get tours of the old Opera House (think Phantom...).
You can get a ride on the Baton Mouche, ride up and down the Seine river seeing the sights. there is also a ride that goes up the small side canal which tunnels under the streets for a ways.
Hop the bus or subway to Pere Lachaise cemetary in the northeast of Paris. A lot of famous people are buried there, but the one to see is the grave of Jim Morrison from the Doors.
There's a whole raft of seconary art galleries - The Orangerie (the water liliy collection of Monet); the Rodin museum (think Thinker); The Orsay (more modernistic stuff, the last 150 years); The tomb of Napolean, and behind it the Army museum.
if you're in the Montparnasse area, you can also climb to top on Sacre Coeur church; although it's a bit of a rip, it looks like one of those ancient medieval churches, but was built in the 1800's. You take a funicular (tilted train) up the side of the mount.
if you want to spend the whole day, there's a side trip by train to Versailles, the huge palace of the last king.
da Vinci code nuts also like to follow some of the scenes from the movie; the newer cathedral with the Rose Line in it, Saint-Sulpice. Apparently the best time to visit is when the organist is giving a concert. Sometimes you can go right up and watch him play.