Help with my first (simple) Z-transform
First of all, it's important to understand that there is no single best way to transform a continuous-time system to a discrete-time system. The method you're using is called backward Euler method, and it is defined by the mapping

$$s\leftarrow\frac{1-z^{-1}}{T}\tag{1}$$

Note that in $(1)$ you scale by $1/T$, where $T$ is the sampling interval (i.e., $1/T$ is the sampling frequency). Eq. $(1)$ just means that continuous-time differentiation (multiplication by $s$) is approximated by a first-order backward difference ($z^{-1}$ corresponds to a delay by $1$ sample). This and other methods for converting continuous-time systems to discrete-time systems are discussed in this answer.

Here is an example where I chose an arbitrary frequency range of interest $f\in[0,10]$, and an appropriate sampling frequency $1/T=20$. The figure below shows the frequency response of the continuous-time system (in blue), and the frequency response of the discrete-time system (in green), which was transformed from the continuous-time system according to $(1)$.