The right way to approach z transform? I am a student learning dsp. I like the subject. I could understand the discrete time signals. When I move into z transform. I could not understand it. 

Z transform is the mapping from discrete signal to zplane to make frequency analysis easy. It would be great if users give some tips that could help me to clarify various views on z-tranform on discrete signals. 

Text is here to explain about broad answers on z-tranform. I am just looking forward in some tips that can help me understand and approach z-transform in text. Just some lines about what you think about z-tranform will surely help. Thank you mate.
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.