Here is an excellent site on intarsia.
It looks like the intarsia that is on the hat is mainly the mouth, right? You'd knit with the brown up to the mouth, knit with white, and then, with another strand, knit with the brown again. The site above has good directions on changing colors without getting a hole. Basically if you hold the color you're no longer using over to the left and then start with the color your switching to, the yarns will twist automatically. Have you looked at Amy's video on intarsia in the Advanced section?
The brim of the hat is worked back and forth, since you can't really do this kind of intarsia in the round. After the color changes are done, then you can switch to circulars.