Take a walking tour of the city - One of the things I like the most about Paris is that it is a walking city. There are tons of neighborhoods to get lost in and lots of shops to stumble upon, not to mention the cafes. Many streets and neighborhoods have passageways between streets that you probably wouldn't find on a map. It's those kinds of things that make Paris so charming. Also, the Metro (subway) is super convenient once you get used to it (but it's not free).
Learn something new at a the Musée Carnavalet - I'm not one for museums and actually I have never been inside the Louvre, which is free every first Sunday of the month. I have been to other Paris museums though and all of them are wonderful.
Hear live music and get a bird's eye view of the city at Sacré-Coeur - One of the things you have to do in Paris is go to area/neighborhood of Montmartre and see the Sacre-Coeur. It will be the gigantic white church that looks like a wedding cake up on the hill. It's as obvious as the Eiffel Tower in that you can see if from just about anywhere. It was in the movie Amelie. This area is one of my favorite in all of Paris even though it's very touristy. There are artists of all kinds there and tons of cafes and shops. One of my favorite memories of being up there is having a Grand Marnier and sugar crepe right on the street. Also the famous Moulin Rouge is not too far, if that is appealing to you.
Do the Avenue des Champs-Èlysées stroll - This is the famous street and shopping area that leads directly to the Arc de Triomphe from Place de la Concorde, which is just west of the Louvre. It is definitely worth walking along even if you don't buy anything.
Get spiritual at the Notre Dame Cathedral - This gothic cathedral is pretty awesome and if you know anything about the history and architecture, you will be very impressed. Also, just a short walk across the bridge Pont d'Arcole is the Hotel de Ville (City Hall) which should look really familiar if you have been to the Las Vegas Paris Hotel.
Spend time outdoors in the city's many parks and gardens
Picnic at the Eiffel Tower - I have never been up the Eiffel Tower in Paris either, but the gardens and area around it are just magnificent with people just hanging out. One of the things I appreciated when I was at the Eiffel Tower is how old it is and looks. It's very impressive.
Take in the music scene - We were in Paris one summer during their national music festival and there are venues and gigs all of the country. I think Lenny Cravitz was headlining that year at Place de la Bastille.
Sample wines at Taillevent
Nosh on couscous
Article: 10 free things to do in Paris, France