Hardwood is acquired by breaking Tree Stumps or Old Logs which need a Copper Axe and Steel Axe respectively or greater to be broken. You will usually get 2 Hardwood per stump, and 8 for an old log.
There is also a very small chance of getting Hardwood from the chests and barrels that are in the Mine. You can also occasionally find Hardwood from the Travelling Merchant.
Many projects require the use of Hardwood, and often there is not enough of it around. South-West of your farm is a forest. In the top corner of the forest, past the big tree, there's an Old Log that requires a Steel Axe to break. Go past the log and you'll find yourself in a secluded area which contains multiple stumps that give around 2 Hardwood each. This area replenishes the stumps everyday so you'll never run out. This is a good method for those struggling to find Hardwood.
Location of Secret Woods where Hardwood can be found.
The Secret Woods.
Location of additional Hardwood.
Hardwood sells for 22g.