If you remain in the USA within the "touristic period" and the training is less than 18 hours per week you don't need a student VISA. As far as I know.
Don't bet on this. You need an M-1 visa if your *primary purpose* for your visit is study.
For vocational studies, the INS interprets this as "more than 18 hours per week" but it has never been officially confirmed of whether this 18 hours period also applies to flight training and if so, what is counted under these 18 hours: flight time, pre- and post briefings, flight prep, self study or anything else.
Want to do flight training on a tourist visa (or under the VWP program), you'd better make damn sure you are indeed legal. Flight training without a proper visa status may get you expelled, fined or jailed, may see your TSA permision revoked for a long time, and may have repercussions for the school itself too.
Also see the link that isi3000 posted. Good discussion on this at the end.