Unlike Chainsaws, the salvaged axe will require players to strike trees manually to bring them down.
There is a chance of finding a salvaged axe from most mid-to-high-tier crates or as loot from most scientists. It is relatively low on the tier 2 workbench tech tree, so learning and crafting them early is also an option.
It is quite normal to use a salvaged axe as a melee weapon, sometimes going so far as to use it as a soft-side raiding tool.
You can take several steps to improve your axe-wielding gameplay when farming. Combine your axing with drinking a wood-gathering tea. Doing this just once with a Pure Wood Tea will likely net you enough wood to last until wipe.