The Boardwalk is closest to MGM by foot, at the path is on the Boardwalk side of the canal. Swan and Dolphin are almost the same, the only difference is you need to cross the canal by bridge, before turning right. YC and BC are a bit longer still, as you need to go around Crescent Lake.
I was at the Boardwalk Inn January 5-9, and from my room it took 11 minutes to walk to the main gate of the MGM. I was alone and do walk at a steady clip, but it was very quick. To minimize the distance, walk out the front door of the Boardwalk, and stay to the right of the auto/taxi circle. Walk down the hill toward the parking lots, and when you get to the lot on the right, turn right. Walk between the tennis courts and the Boardwalk Villas, then turn left onto the path.
Good luck and have fun.