The first place you find a DMR is during the second mission, "Winter Contingency". Once you get out of the Falcon, you will be told to "Retreat into the Relay Outpost" after defending it. After the cutscene, you will be put into a dark corridor. Immediately turn to your right, and you will see two DMR rifles on the gun rack on the wall.
You start the third mission, "ONI: Sword Base" with a DMR also.
DMR - M392: The DMR is a single shot rifle(semi-automatic) with a scope on top that has a 3x zoom. To use the scope, press . The easiest enemies to kill are the grunts since they have no over shield or shield on them. Aim for the head for easy kills.