To use an Axe in New Horizons, equip it from your pockets or your quick-tool menu so that your character is holding it. Then simply walk up to the tree you want to chop up and press A to swing at it.
The Switch Axe is one of 14 weapons available in Monster Hunter Rise. This page covers every tree available for the Switch Axe. These are just the final forms available for each of the weapons ...