I normally get fangbeasts when I harvest sugar cane and tend sheep or chickens. Skunkupines appear when I tend pepe (the pet skunkupine), harvest wheat, and tend flowers.
I'm still working on finding skunkupines and fangbeasts, but I'm in no hurry. If I get one while doing normal things around my kingdom, yay. If not, no big deal. Since there isn't a big prize at the end of finishing this goal line, I figured I'll just take my time and enjoy the story.