Obtaining Sticks and Logs
To obtain sticks you will need to find branchy leaves.There are two type of leaves in the game, regular and branchy. Regular leaves have an extremely low chance of dropping sticks and saplings. You can break branchy leaves easily with your hands which is guaranteed to yield you sticks. However, if you want more sticks, you should use shears instead.
Once you have made an axe, you can fell an entire tree by chopping the bottom most log. This will break the leaves too, however, your stick and sapling yields will be lower than if you use your shears on the leaves.
Here is a "Sticks and Logs tutorial" or "how to make basic tools and fell trees" video guide by SmileOnSpeedDial