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According to Fraraday's Law, the magnitude of the induced emg, hence induced current, is directly proportional to the rate of change of magnetic flux, i.e. rate in which the magnetic field lines cut the solenoid and vice versa.
To get a higher induced current:
1) Increase the number of turns in the solenoid per unit length
2) Use a stronger magnet to increase the magnetic field strength
3) Insert the magnet in and out of solenoid at faster speed
To reverse the direction of the induced current:
1) Reverse the winding of the turns on the solenoid
2) Reverse the pole of the magnet
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