There are two ways to get furniture paint. The basic colors can be obtained by clicking "furniture paint" spawns in Oriel and Pernon. The special colors can be obtained from a guestbloom that you can buy from a village merchant. You must have purchased a home (studio doesn't count) in Oriel or Pernon in order to have access to the merchant that sells this guestbloom.
Here are all of the furniture dye colors you can get:
Here are all the locations of collectable furniture dye spawn points in Oriel:
Here are all the locations of collectable furniture dye spawn points in Pernon: