It's easy guys. The PM hands the thrust levers between V1 and Flaps Up in case of takeoff or between Go Around call and Flaps Up in case of go around. During all other times the PF hands the thrust levers.
To accelerate during the climb, simply follow the FD, it will command a slight pitch nose down to around 10-12,5 degrees and as you retract flaps the aircraft accelerates nicely. At 1000 ft to go, simply lower the nose to 10 degrees to reduce the rate of climb and thrust levers to 80% N1. It will make the level off smoother and controllable.