On this page of the Valheim guide, you will learn where to get the Core Wood.
There are a total of 3 different types of wood in the game. Each is obtained from a different type of tree. Obviously, the easiest to get is the basic type, as it drops from every log. The other types have to be obtained from a specific tree.
You can get core wood by cutting down pine trees. These are tall, coniferous trees that you can find in the Black Forest. This biome appears frequently, and it's always on the starting map, so you won't have any trouble getting to it. The density and frequency of pines within this biome is random. In one spot, you may find only a single pine tree, while in others you may come across forests made of mostly pines.
You don't need advanced tools to cut pine - a basic stone axe can handle this type of wood. This means that you can start cutting them down right from the beginning. Keep in mind, however, that this type of tree cannot be stacked. This means that you need a chest to store larger stocks of this resource.