Forest & Cave
Forest is where the wood and stone materials are. To get the resources, you need to send one (or four if you have unlocked all four Areas) of your Pets to the Forest.
To access the Forest/Cave, click/tap on this area,
Then you get this kind of screen (both are pretty similar),
There are 4 Areas that you can send your Pets to. Depending on the level, each Area (or most players called it slot) drops a different tier of materials. Here are the drops from all Area
Forest
Cave
Warning! Do not select the Pet first! Make sure to select the duration first, because it will use the pet immediately if you tap/click right away!
And the last one, using Worker Vocation while gathering the resources here gives you another special drop that is exclusive to Workers!