If you have a CPL you go to the store and buy it... they will provide the paperwork and tell you where to take it. NOTE you do not take the gun in for an inspection any more. IF from a private party you will use this form www.michigan.gov/documents/ri-060_6454_7.pdf
If you do not have a CPL you will need to get a Purchase Permit from your local police or Sheriffs dept. if you are rural. You will need to get it notarized before you go to purchase the pistol. Some places will do it free some for a fee however you can have it done anywhere that has a notary.