You can join the Army as a MP right off there is no time in service requirement needed, its a entry level MOS if you qualify for it after you take the ASVAB then you can pick it min score is 95 last time I checked, but its been a while.
Basic and AIT are at the same place are are back to back its called OSUT at Fort Leonardwood, MO.
Don't expect to being doing police work 24/7 either. You will be deployed no matter what, MPs are one of the most deployed MOS's the Army has. You will either go to a BCT and be with a MP Platoon and do whatever the BDE wants you to do, or be apart of a MP Company in a MP Battalion and to though cycles of Admin, Field/train to deploy, and MP road duties. MP Company's while deployed can be expected to do almost anything, from training Iraqi/Afgan Police to Combat Patrols same goes for the BCT MP's.
We do have Law and Order Detachments, they mainly only let experienced MP's go to these units and ones that have 1 or 2 or more deployments under their belts. They have sections ie: SRT, MPI, TAI, K9/MWD, Physical Sec., MP Desk, and VRO. You get sent to the schools for those jobs and get the ASI's (special skills). These are the guys that do police work 24/7.