In order to cut down trees in My Time At Portia, you will need an axe. Creating one will cost you 13 x Stones and 13 x Wood, which is a bit tricky since you can't really cut down trees to get wood before you get your axe. Anyway, there's another source to getting the wood needed to craft the axe, which is found by looking to the left of the entrance to Peach Plaza. There is a grassy area there where you can collect both Wood and Stone without any issues.
It should not take too long to collect 13 of each, and once you have enough, head back to your worktable right outside of your workshop building and craft the axe. Select the axe and confirm that you really want to craft it, and it will be ready to be used after that. And that's all you need to do to get your axe in My Time At Portia!