Almost everywhere in Paris is a nice area to walk around.
Try visiting the chapel of Sainte-Chapelle, at the Palais de Justice, to see the most beautiful stained glass windows in Europe. Because it is part of the Palais de Justice, it has limited hours.
A great way to experience the tour Eiffel for the fist time is to start at Trocadero, and walk down through the park toward it.
As the locals do, find a neighborhood café on an interesting street corner, and savor your coffee at a table and experience the passing view.
Paris is great for everything edible. If you like food, it is good not to live there, because everything is so tempting. It has the best pastry in the word, and patisseries everywhere. Try Stohrers, at 51 rue Montorgueil (M:Les Halles), for the best financiers. Rue Montorgueil is a great food market street, with some good fromageries. If you ask for camembert, the proprietor will ask you if want one that is to be eaten now, this evening, tomorrow or later, so that he can select the cheese of the best ripeness. You may be able to try vacherin, a winter cheese (although it may not be late enough in the year yet.)
Very near by is L’eglise Sainte Eustache – a very beautiful church which is not very touristy - at the beginning of rue Montorgueil, close to Les Halles.
For a gay restaurant (hetero friendly) try Aux Trois Petits Cochons (also close to rue Montorgueil).
If you like Monet, try visiting the Musee Marmottan near the Bois de Boulogne. It may be (it used to be) very uncrowded – everyone was going to the bigger museums.
On a rainy morning get up very early, and walk around the Ile de la Cite, or anywhere along the Seine – great for black and white photography.
Walk – almost anywhere. Paris is for walking. Explore the Jardins de Luxembourg. Walk through the Marais – the gay area.
Go to a tourist office and inquire about local concerts – there are usually musical concerts – often in churches – that are free or very inexpensive.