If you have a lot of time take public buses around the city - can get busy but much cheaper than tourist buses. If you're short of time use the Metro or RER.
I would recommend all the places AfricaResource did plus Invalides, Palais de Luxembourg, Trocadero, Buttes Chaumont (nice park for a wander), Parc Monceau, Republique, Bastille, La Defense (if you like modern stuff), Opera and Place Vendome (if you have an unlimited credit card!!).
The best thing to do is just to wander and you will suddenly come across beautiful views or take Metro lines 2 or 6 which are partially overground. If you like trains take a trip on a TGV to somewhere - they travel at around 200mph!