I know how slip and flux angle individually affect torque of induction motors but I can't understand the relationship between slip and flux angle in terms of torque. Do they both have different effect on torque or one cause the other (linked to each other)?
It is really load torque that effects slip angle and torque not the other way around. An induction motor has very little slip and a very small torque angle when it is not connected to a load. When a load is connected, increasing the load causes slip and torque angle to increase in proportion to torque. In the normal operating range, the increase is fairly linear. A VFD that is properly designed and working properly will keep the motor operating such that the relationship between torque and slip or torque angle is as linear as possible.
Dynamically, if you want the load to accelerate, you increase the frequency and that increases the slip creating torque to accelerate the load inertia. The torque angle will also increase. The VFD should limit the torque to an acceptable level. It could do that by determining either slip or torque angle and limiting the rate of frequency increases accordingly. There are several control strategies that could be employed to accomplish the same result.