Why is there so much movement of the sidestick during this takeoff?
One of the unique aspects of Airbus fly-by-wire aircraft is that the sidesticks provide NO feedback. Because of this, the planes have an automatic trim system. Once the aircraft is pitched to a desired attitude, the pilot will then release the control stick and the aircraft will continue to fly at the selected attitude. 

Correspondingly, if a pilot desires to reduce the pitch, they must push the stick forward to do so. In an Airbus (unlike most other aircraft), a pilot doesn't manipulate trim at all (except in non-normal situations). 

As an aside, in the normal flight regime, Airbuses provide automatic protections from unsafe attitudes. Thus, if a pilot desires maximum rate of climb (that is possible in the current configuration), he can simply pull the sidestick all the way back. The computer will automatically keep the aircraft from exceeding maximum load factor and/or critical angle-of-attack.

To answer the other part of your question, the lateral stick movement is likely due to turbulence. Control inputs in large aircraft tend to be more pronounced than smaller aircraft (due to slower aircraft reaction times).