First of all, as a Handyman you will be practicing a crafting profession. Previously a permit was needed for this profession and you were unable to use harvesting professions; however that is not the case anymore due to an update.
Handymen make furniture, furnishings and crafting workshops for Haven Bags. You will need to gather wood and craft planks.
Once you have gathered all these ingredients, you will need to find the nearest handyman's workshop to create your items using one of the wood lathes available.
Warning: The following text is a spoiler, select it to read:
1) The handyman test has begun. What materials does a handyman use? --> Wood and planks.
2) Which profession can supply these ingredients? --> Lumberjack.
3) Very good. But what does a handyman make with all that? Display windows, furniture, and crafting machines for Haven Bags.
4) We're on the home straight now: which equipment do you use to make all that? --> A wood lathe.
Handymans use a Wood Lathe for planing their items.
For locations of Wood Lathes, see the main article: Wood Lathe