Since you are new to Control Theory as you say and also new to the forum, I will give you some hints on what you are asking. However, in the future you should initially do your own search and if it is essential ask here providing what you have done so far and what exactly you didn't understand and confuses you. I encourage you to read also this thread on how to ask good questions:
A controller is a structure which a designer implements in order to be able to force certain behavior to a specific physical system. The PID-Controllers are probably the most known of these structures since they are widely used in the industry due to their relatively easy design procedure (although I believe that there are numerous aspects to be taken into consideration when designing and tuning a PID controller) and because they can even be tuned (selection of the $K_p,K_d,K_i$ gains) without the knowledge of the mathematical model whih describes the real system. However, there is a vast majority of controllers which an engineer can choose to implement like lead-lag compensators, state feedback controllers (full state or dynamic state), linear quadratic regulator and so many more. To conclude, yes there are many other controllers which you can choose to implement.
- The aspect of: "Better" Controller
Sort Answer: There isn't one controller to rule them all.
Long Answer: The choice of the final controller to be implemented and deployed to a microprocessor in order to control a physical system is not at all straightforward. The engineer has to take into consideration the operating conditions of the system, the physical characteristics (sensors, actuators) and their limits, the mechanical structure of the system, the disturbances, the measurement's noise in order to be able first of all to understand the system. Moreover, he has to decide if he wants robust controllers (better to deal with unknown system dynamics, unexpected environmental conditions, intense nonlinear systems approximated by simple linear ones) or optimal controllers (achieve better steady state and transient response characteristics). After these thoughts and maybe many more he implements the controllers and checks which one fits better to the specific system with respect to the steady state and transient response characteristics which need to be achieved.
So, in order to find a controller which produces better transient response for your system you need to design one and test it directly to the system you are coping with. Nobody can tell you by a guess which controller is better than the PID. You may even conclude that you can't implement a controller that achieves better transient response characteristics. In other words you need to perform you own mini research regarding the specific system you are studying in order to come up with what you are asked to.