Most museums and galleries close on either Monday or Tuesday. In general, state-run museums (Louvre, Centre Pompidou, Rodin, Cluny, etc) close on Tuesday and municipal ones on Monday (Carnavalet, Petit Palais, Bourdelle, etc), with a few exceptions (state Musée d'Orsay closes on Monday) and independently run museums and galleries may have close one or two days a week, or not at all – so check first: it's tragic to see all those people milling around outside the Louvre on the day it's closed.
You may find queues shorter in the evening for those places with late night opening.
Paris's municipal museums have free entry, though generally not for temporary exhibitions. National museums are free for all under 18s and all European Union nationals under 26.
If you're planning very intensive sightseeing, the Paris Museums Pass (www.pasismuseumpass.com), which gives free entry to around 60 museums and monuments in Paris and the Paris region – see the website for the full list – can be good deal, especially as it often allows you to queue jump at ticket desks. It is available for two (32 euros), four (48 euros) or six (64 euros) days, but generally does not include temporary exhibitions, and can be bought at tourist offices, selected museums and monuments and branches of Fnac.
How I’ve picked my things to do:
As well as the mega museums like the Louvre, d'Orsay and the Centre Pompidou and must-see icons like the Eiffel Tower and Notre-Dame, I've also chosen some of Paris's smaller specialist museums, such as the Musée Gustave Moreau, which has the personal appeal of the artist's own house and studio, lesser known churches like the beautiful Eglise St-Etienne-du-Mont that forms part of Parisian history and some of the squares and parks that are an intrinsic part of the urban landscape and places where Parisians themselves go to relax. There's enough in Paris to keep you occupied for months, these are just a starter.