Relation between activity, intensity, count rate and so on considering ionizing radiation

I’m currently doing a project on waste products of nuclear power. To narrow it down, I chose to focus on gamma rays in particular. However, I generally find it hard to see the relation between intensity, activity, count rate, energy level and so on. For instance, I know that intensity is directly proportional to count rate, but why is that? Currently, I’m working on a basic experiment where i choose different materials to expose to gamma rays and see how the count rate decreases as a function of increasing material thickness. However, I think I need some more theory on the above mentioned concepts (activity, intensity etc.) to apply it for something useful.

I was hoping you smart people in here could help me get a better overview of everything. Thanks in advance