After they’ve done it, the axe recipe will be available to other players.
Go talk to Tom Nook.
He’ll give you the Flimsy Net and Flimsy Fishing Rod.

Use those tools to capture to bugs andcatch some fish.
When you’ve got a few of each, go to back and talk to Tom Nook again.
Donate the fish and bugs to him and he’ll give you the recipe for the Flimsy Axe.

Collect five tree branches and two stones from the ground on your island.
When you’ve got those, find a workbench, choose the Flimsy Axe recipes, and craft it.
It will break after about 40 uses, and you’ll need a new one.

Don’t settle for another Flimsy Axe, though.
Perform various tasks around your island until you’ve amassed 3,000 Nook Miles.
When you’ve got 3,000 Nook Miles, go to the Nook Stop terminal and selectRedeem Nook Miles.

From there, buy thePretty Good Tools Recipe.
This contains recipes for upgraded axes and other tools.
Collect the materials you’ll need to upgrade your axe.

Check out the next section for the kinds of available axes and what you’ll need to make them.
You don’t have to gather materials and craft a new axe every time you need one.
After an axe breaks, you could buy a new one if you have enough bells.

Of course, you could still craft them from raw materials if you prefer that.
Here’s how to get each recipe, what materials they need, and how long each lasts.