The Basic Axe comes through task progression and arrives fairly late in the demo. Keep completing available tasks if it has not appeared yet.
Quick answer
The Basic Axe comes through task progression and appears fairly late in the demo. Not having it early does not mean you have missed it.
How to get the Basic Axe
Keep progressing through the tasks currently available to you. The Basic Axe is awarded through a task, so advancing those objectives is the way to unlock it.
Did you miss the Axe?
The Basic Axe arrives fairly late in the demo. If it has not appeared yet, continue working through your tasks rather than assuming an early opportunity has passed you by.
What to do for wood before getting the Axe
Starting on day 5 after receiving mail, a wood-buying option becomes available. Wood can also appear around Town, Town Outskirts, Mall Outside, and Lakefront.
Wood availability is randomly generated, so the amount available can vary.
How tool upgrades help
Tool upgrades reduce the number of swings needed to break equivalent resources. One upgrade generally halves the swings needed for the same resource.
Fewer swings also mean fewer chances for negative outcomes triggered by swinging, including damage, curses, or battles.
Do tools need to be equipped?
Tools do not need to be equipped. Keep the appropriate tool in your inventory to use it.