How did Fats learn to play? His father taught him to play the harmonium when he was five, and he took lessons from a local teacher (apparently unknown now), but he taught himself (before the age of fifteen) by the following method, as described by Paul Posnak: "By 1916, the great eastern rag and stride pianists Eubie Blake, Lucky Roberts, and James P. Johnson had all cut piano rolls. By slowing down the roll, tracing the key movement with their fingers, then releasing the locking mechanism and practicing the chords they had just learned, young pianists like Fats Waller, Duke Ellington and many others taught themselves the styles and techniques of these master pianists." At the age of fifteen (that would have been in 1919) Fats was playing at the Lincoln Theater in Harlem for $23 a week, accompanying the slient films and entertaining the audience in the breaks. When his mother died, and his father (who was a "lay preacher") disapproved of jazz, he moved in with the parents of his pianist friend Russell Brooks, taking over Russell's room (available because Russell had just married and moved out). He practiced on their player piano, and this was where he got access to a player piano to reverse-engineer the rolls. The Brooks family had many piano rolls by James P. Johnson. At night, Fats went out to the all-night parties known as "rent shouts". There he met and got to know Johnson. Paul Posnak says: "When he first heard Fats play, he said to his wife, 'I know I can teach that boy.' Within weeks of Fats' move into the Brooks household, he was studying with James P., eating Mrs. Johnson's food, and practicing on their piano until Mrs. Johnson made him go home at 3 or 4 am. James P. taught Fats to play stride. He also taught him classical keyboard technique, principles of tone production, transposition, and harmonic and motivic development."
Apparently Fats also had some instruction in his late teens from "Willie The Lion" Smith. I haven't yet found the details of this relationship and instruction. Willie the Lion is considered one of the three "masters of stride piano", along with Fats and James Johnson; he is less well-known today, perhaps because he recorded very little compared to the other two masters. Fats Waller, on the other hand, recorded more or less continuously from 1922 until his death in 1943.