There are two aspects to cricket batting which I think club players like us should work on: 1) Physical strength and conditioning; 2) Technique
I was in a similar situation as yours and too slow for quicks. Physical strength helps a lot. You should look into high intensity interval training to get used to quick body response to get into the right position for different shots.
@uknsaunders has provided a great example of how to use weights to train for specific shots. You should try it. If you watch videos of Virat Kohli training in the gym, you will see that he is using barbells to develop quick body reactions. It is just one aspect of his training regimen but does highlight this major point about cricket batting.
My observation is that inorder to play a cricket shot correctly, not only do you need to understand the (physical) mechanics of playing the shot but also the conditions (type of deliveries) to play the shot. I love playing the pull shot but there are deliveries that I leave because I can't pull them. Trick to playing the pull shot is to get in position early by gauging the length early. This is a skill that require a lot of practice. Once you are in position, you can pull the ball without problem. You need that extract micro-second to get bat to ball and early back-foot position will give you that. It is all about muscle memory and reaction time; you should train for that. Bat weight has nothing to do with it. I have pulled real quicks with heavy bats because I got in position quickly. Bat weight had nothing to do with it. Start slow. Use tennis balls or low speed Bola machines. Gradually, increase the Bola speed. Note of caution: Bola balls can do serious harm at high speeds - I ended in an ER thinking my arm was broken when I missed by pull (over powered) and got hit on my right forearm.
Let us know how it goes.