Anyone who's been to Paris and who claims to know anything about art will have visited the Louvre, the Pompidou Centre and the Orsay, I believe. But to ignore the other galleries and museums is to miss out on much of the richness and vibrancy of Paris.
I have explored two main clusters, though naturally there are many more out there. The first of them is in Saint-Germain-des-Pres, based around Rue Furstemberg, quite close to the Seine and the Orsay. Its major anchor is the Musée EugeneDelacroix, devoted to one of the foremost French Romantic artists. Once his final apartment, it now contains both memorabilia and letters, as well as examples of his works from throughout his life (there is an admission charge, but it is free for 16 - 25 year old residents of the EU, and is open daily bar Tuesday.)
What really adds to this location is the number of small, private galleries you can take a look around. Simply walk down Rue Furstemberg and Rue Jacob and you will find dozens of them, of all different genres. It is slightly surreal, the blend of the more traditional and the distinctly modern on offer - post-modernist experiments to Pre-Columbian sculptures (the latter which were on show at the Galleries Furstemberg in February.) The area also boasts an impressive array of antique vendors, also providing ample window-shopping for those with the time.
The other zone, in the historic Marais of the 3eme Arrondissement, is based around Rue Thorigny, and features a number of both museums and galleries. Although the National Picasso Museum there is closed until summer, a plethora of galleries offer works to enjoy and buy (many are open to the public for free, such as Roi Doré, which showcases Polish artists.)

On the other hand, there is always Musée Cognaq-Jay. Set in the Hotel Donon, a 16th Century 'hotel particulier' (a sort of town house), it houses an impressive collection of 18th Century art-works in fabulous settings. Eschewing the neutral styling of other museums, Cognacq-Jay works the pictures and sculptures in with ornaments, time-pieces and chandeliers to give it a fantastic period feel, adding gravitas to the whole experience (currently on display are works by sculptor Dalou.) It is also just a short walk from the Musée Carnavalet, which has the History of Paris for those who are less inclined towards art, or would just like something different.

Of course there are many other galleries and art museums in Paris - these are just suggestions to hopefully pull people away from the 'Big Three'. Because after you've caught the Museum Bug, who knows when you'll finish?