The right way to approach z transform?
Discrete-time signals are generally written in the time domain as $x[n]$ where $n$ is an integer.

We want to be able to see how discrete-time systems can be used to modify these signals:
$$
y[n] = \sum_{m=1}^M a_k y[n-m] + \sum_{p=0}^P b_p x[n-p]
$$

The above equation is a linear, constant coefficient difference equation.

Such equations can be hard to manipulate for a specific $x$ to figure out what $y$ is.

The $z$ operator is used to attempt to simplify the analysis of such systems. $z$ is the forward-shift operator: it shifts a signal forward by one time instant. $z^{-1}$ is the backward-shift operator: it shifts a signal backward in time by one time instant.

Using this operator, we can rewrite the equation as:
$$
Y(z) = Y(z) \sum_{m=1}^M a_k z^{-m}  + X(z) \sum_{p=0}^P b_p z^{-p}
$$
which, upon rearrangement, we can get:

$$
Y(z)/X(z) = \frac{\displaystyle\sum_{p=0}^P b_p z^{-p}}{\displaystyle1 -  \sum_{m=1}^M a_k z^{-m}}
$$

which is now a discrete-time system that we can analyze in the $z$ domain, rather than the time domain.

One thing this buys us is the ability to figure out what happens when many discrete-time systems are put together: we just multiply the discrete-time system descriptions (transfer functions).  Doing the same thing in the time domain is doable, but very tedious.