There are 5 built-in games you can play through the Game menu.
The first two, Climb, and Tug-of-war, award skill points to a random category.
The other three award skill points to specific categories.
Apple = funny (smiling face icon)
Shape = gorgeous (star burst icon)
Manhole = spiritual (fist icon)
Note: the number of games you can see in the Game menu depends on your tama character's age. Babies = 1, Toddlers = 2, Teen & older = 5.