Left Turning Effect
 

The plane moves to the left.
So we are supposed to compensate for it to the right in terms of roll and yaw?

This can easily turn into an hour worth of ground school.
Left turning tendencies are (largely) caused by a clockwise turning propeller. A prop that turns counter clockwise creates right turning tendencies.
On a simple airplane like a Piper Arrow the engine mounts are angled slightly toward the right.

https://cimg1.ibsrv.net/gimg/pprune....d0368af860.jpg
The horizontal stabilizer is angled slightly to the left. This is supposed to cancel out any turning tendencies in cruise flight.
During high power settings such as used during take off some right rudder is still required to keep things going straight.
With high power engines (think P-51 Mustang) full power is only applied well into the take off roll when sufficient airflow over the tail is achieved. Aileron is used to stop a torque roll in high power aircraft.

http://www.kathrynsreport.com/2016/0...ng-n1451d.html