Chopin arrived in Paris in 1831 at the age of 21; he would never return to Poland. While in Paris, he came into close circles and developed relationship with the likes of Berlioz, Liszt, Delacroix, Vigny, Mickiewicz. It is here that he met George Sand and began their famous relationship.
While in Paris, you can easily visit many areas around the city where Chopin left his mark. You can visit the apartment where he spent most of his years in Paris at the foot of the Montmartre hill, located in the 9th arr. at 80 rue Taitbout. This is the inner courtyard of his apartment complex:
George Sand also had a neighboring apartment with her family within the complex, so that they could always be near each other and she could keep an eye on him with his declining health. (Both of their apartments are marked with small plaques).
Continuing near his apartment in the Square d'Orleans, you should visit the Musée de la Vie Romantique (16 rue Chaptal). I think it's a great museum even if you're not there solely for Chopin! Originally, this museum was the home of the Dutch painter Ary Scheffer; him and his daughter often hosted Friday night soirees, where Romantics of the time would gather, share their work, and discuss. Romanticism boomed in Europe in the early and mid-1800s, with a large focus in Paris. Romanticism was characterized by its emphasis on emotion and individualism. in reaction partially to the Industrial Revolution and the aristocracy of the Age of Enlightenment, it was often associated with liberalism and radicalism.
Within the museum, you can see writings by George Sand and memorabilia, the original casting of Chopin's hand (which I'm still amazed at his joint flexibility...his hand seemed to be about the same size as mine, but he could reach 2 or 3 keys further than me. Depends on the day :)) And there are, of course, many other sculptures and paintings from this period.
If you're not much into the Romantic period, or museums, this courtyard is a beautiful place to have a tea and enjoy the surroundings.
(Discounted rates into the museum for students).
While in Montmartre, don't forget to walk up the hill and explore the ivy-covered neighborhood, where Chopin spent many of his days.
Also in Paris, you can visit the Polish Library, the first non-French library to be opened in Paris. There is a Chopin salon (shown in the picture below), where you can see original scores, letters, and drawings of Chopin. Besides the Chopin salon, it houses the Adam Mickiwicz Museum and the Biegas art collection. If you can read French, there is a good room on the history of Polish immigrants in Paris. It is cited on the UNESCO Memory of the World register as "an institution unique of its kind".
(If you are a student of the EU, between 18-25, you can enter for free).
Lastly on our Chopin tour is his famous tour within the Cimitiere du Pere Lachaise. Sick for many decades, Chopin's health seriously began to decline in the 1840s. His relationship with Sand also severely declined, with her becoming more of a nurse rather than a lover. (She often called him her "third child"). She published an allegorical book in 1847, Lucrezia Floriani, whose main characters, a rich actress and a prince with weak health, seemed to paint their relationship. it showed Chopin in a very unfavorable light, and upset many of his friends -- Liszt completely broke off relations with her after the publication.
As he was dying, he planned his funeral and the arrangements. He was terrified of being buried alive, so he requested that his body be opened and his heart returned to Warsaw. His heart was smuggled and buried in a pillar within the Holy Cross Church (Kosciol Swietego Krzyza) in Warsaw.
Chopin’s funeral was celebrated in the Church of the Madeleine on October 30, 1849, after a two week delay. On his death bed, Chopin murmured, "Play Mozart in memory of me -- and I will hear you". While making his arrangements, he instructed for Mozart's Requiem to be sung at his fineral. This caused the delay because at the time, the Church had never allowed for female singers to be a part of the choir. In the end, they relented, but only on the condition of the female singers to stand behind a black curtain. (Look at the Romantic liberalism, even after death!).