Does a time-series have to be stationary before you calculate a z score or t score? It's been a long time since basic statistics. I have a financial time-series that exhibits exponential growth. 
Before I standardize, do I have to make the time-series stationary?
Before I standardize, do I have to normalize?
 A: Can you show us your particular dataset? If you are interested in inferring things about $\mu$, the parameter that describes the average change, then you might need to assume more than stationarity, maybe some assumptions about how autocorrelated the series is. In general, you don't want the autocorrelation to not die out. If it's a financial time series, it probably won't be a problem. Also, estimates of $\mu$ are less noisy than z-scores.
Here are a couple technical results:


*

*If a time series is stationary with mean and covariance functions $\mu(\cdot)$ and $\gamma(\cdot)$, then $\text{Var}(\bar{X}_n) \to 0$ if $\gamma(h) \to 0$ as $h \to \infty$.

*If a time series is stationary with mean and covariance functions $\mu(\cdot)$ and $\gamma(\cdot)$ then $n E[(\bar{X}_n - \mu)^2] \to \sum_{h=-\infty}^{\infty} \gamma(h)$ if $\sum_{h=-\infty}^{\infty}|\gamma(h)| < \infty.$
You probably want the second result. You can see that 
$$
\text{Var}\left[\sqrt{n}(\bar{X} - \mu)\sigma^{-1}\right] = n\sigma^{-2}E[(\bar{X}_n - \mu)^2] \to \sigma^{-2}\sum_{h=-\infty}^{\infty} \gamma(h) < \infty.
$$
If you have a reasonably large dataset, your z-score's variance will be close to some constant number.


*If you want to assume further that, in addition to being stationary, your time series is Gaussian, then 
$$
\sqrt{n}(\bar{X} - \mu)\sigma^{-1} \sim \text{Normal}\left(0,\sum_{|h|<n}\left(1 - \frac{|h|}{n}\right)\gamma(h)\right).
$$
This is exact for any $n$, and it will help you come up with confidence intervals for $\mu$. Similar results are available that show this is approximately true, without assuming normality, as long as $n$ is large.

A: First: (1) I assume that by standardize, you are wanting to know how many standard deviations a particular observation is from the mean;  and (2) I assume that by normalization, you are meaning a transformation to express your data series on a scale of between 0 and 1 - from the minimum to the maximum observation.
For (1): If you standardize, then this tells you how many standard deviations an observation is away from the mean. If the series is non-stationary, the mean could be changing, and if the mean is changing, the variance (and standard deviation) are likely changing too.  So, you would not be able to standardize without considering these effects.
For (2):  I do not see how normalizing would help in this context.
There are many ways to analyze time series data, but you need to provide some further information about both the data set you want to analyze, and what you want to determine from the analysis in order to get more help here.
