Believe it or not, MP are no more stronger or tougher than other soldiers, especially when compared to Rangers or other Special forces. For this reason, MP's do not involve themselves in a fair fight. If they put up their fists, you would put up your baton or pepper/OC spray. If they have a knife/bat you would use your taser or pistol.
MP training is 10 weeks long. On the homefront you will learn paperwork (lets out an unenthusiastic "yay") unarmed self defence, use of the baton, and use of the pistol. More importantly you will learn combat skills such as use of the M19 (which is basically a firearm with a big bullet) the M249 and convoy operations. You will also learn how to handle prisoners which is a little combat and a little bit of garisson.
Being an MP requires a secret clearance, and an alleged "high moral standard" compared to the rest of the army, your criminal record cannot contain dui's or many other crimes. However, if you're like me and are a speed freak. One or two traffic tickets won't be a problem.
You will eventually have to pass the army physical fitness test required for ALL soldiers. If your an 18 year old male this requires you to do 42 push-ups without stopping in 2 minutes, 53 sit-ups and run two miles in 15:45, which is a 7:23 minute mile.
Getting in for me was rather easy. But you came into the army physicaly unfit; Lost 57 pounds before you came into the army and lose 10 more after training, you will find it difficult to keep that weight off for very long once the Drill sergeants stop monitoring your diet.
I have been in the Army National guard for 1.5 years, but pay me no mind because I am still an E-1 (the lowest rank there is), where most of my peers are E-3.