Dog sled drivers use various command words such as "Hike!" and "Mush!" to get dogs to run faster and respond to cues. "Gee!" and "Haw!" are used to direct the dogs' direction on the trail, while "Whoa" means stop and "On by!" instructs the dogs to pass other teams or distractions on the trail. These commands ensure communication between driver and team for a safe and efficient journey.