How do I increase the output voltage of a simple electro magnet generator
By Faraday's Law the voltage is proportional to the number of windings and the rate of the change in the magnetic field. So you can add windings, as you mentioned, move the magnet faster or use a stronger magnet at the same rate to increase the voltage. I would look at http://hyperphysics.phy-astr.gsu.edu/hbase/electric/farlaw.html and http://www.electronics-tutorials.ws/electromagnetism/electromagnetic-induction.html for an explanation of Faraday's Law.

From wikipedia: