What forces cause the centripetal acceleration in circular motion on a merry-go-round? Suppose you stand on a merry-go-round spinning at $f$ revolutions per second and you are $R$ meters from the center. What forces act on you? 
If you were tied to a rope like in Fig 2 you would follow a straight trajectory after cutting the rope (in the absence of gravity) so I figured that your direction of motion is like in Fig 1 - perpendicular to the direction of acceleration. Since the direction of friction is specified by the direction of motion, it must be the opposite direction, right? But this is all wrong and I don't know why.

 A: 
Since the direction of friction is specified by the direction of motion, it must be the opposite direction, right?

Wrong. You can go back to whoever told you that and yell at them. :-P
Seriously though: the direction of friction actually has nothing to do with the direction of motion. One really obvious way to see this is that you can make the direction of motion be anything you want, just by changing how you (the observer) are moving. For example, consider a hockey puck sliding forward on ice. Friction points backward. But then imagine that you're a hockey player skating forward, faster than the puck. Now you see the puck moving backwards (relative to you, of course). Yet friction still points backwards! It wouldn't make sense for the absolute direction of friction to change depending on how you're moving.
What determines the direction of friction is the relative motion between the two surfaces that are experiencing the friction. Specifically:


*

*Static friction acting on an object points opposite to the direction in which the object would slide along the other object if static friction didn't exist.

*Kinetic friction acting on an object points opposite to the direction in which the object is actually sliding along the other object.


That's kind of complicated phrasing, so here's how it works in your example:


*

*Imagine an observer moving along with the "other object". In this case the merry-go-round is the "other object" and you are the "object", so imagine an observer who is just standing on the merry-go-round, affixed to it.

*Imagine that there is no static friction. What would happen? You go flying off the merry-go-round. The fixed observer would see you fly outward.

*Since you would move outward without friction, therefore static friction acts inward.

