Dragging the logs out of the woods is called skidding. Today they have machines called skitters to drag the logs with. Previously they might skid logs with a horse team. I read of one guy that used his pet moose to skid logs.
Waterways have always been the most economical way to transport logs. I expect that your grandfather had a number of short lines with a log dog at each end. He would dog the tail of oe log to the head of another and tow them in one behind the other. He could only tow a few at a time with a skiff. Larger boats would make rafts with bundled logs, if the water was deep enough.
Pet moose? Was it named Babe?
Up until the past 30-50 years lumber and paper were some of the biggest industries in my part of Florida. I don't know how the logs were transported since the terrain here is so flat our rivers have very little current. We used to see trainloads taking pine logs to a papermill in the next county, and we also once had a gunpowder plant. Now, most of what is harvested comes from construction sites for the latest shopping center or housing development and everything gets transported by semi-truck. The papermill is long gone and I can't imagine what use some of the cut trees can have because they can be seriously lacking in diameter.