What frequency bandwidth should an accelerometer operate for measuring punching and kicking movements?
This article talks about 50g and milliseconds for boxing. Based on that, you should maybe start out with the full BW (1kHz'ish). By experiments, you may learn that this can be dialed back later.

BTW: I once connected an ADXL-xxx device to a Logomatic datalogger (both the accelerometer and a flat LiIon battery fitted nicely on the back of the PCB). That proved very useful for similar experiments. Maybe you can do the same thing. I suspect you don't want to use a smartphone for initial experiments...