What are the standard statistical tests to see if data follows exponential or normal distributions? What are the standard statistical tests to see if data follows exponential or normal distributions?
 A: For the exponential distribution, you can use a test called Moran's or
Bartlett's test.  The test statistic $B_n$ involves the sample mean
$\overline{Y}$ as well as the sample mean $\overline{\log Y}$ of the
logged $Y_i$
$$
     B_n = b_n \times \left\{\log \bar{Y} - \overline{\log Y} \right\}
     \qquad b_n = 2n \times  \left\{1+ (n+1)/(6n) \right\}^{-1}
$$
Under the null hypothesis we have approximately  $B_n \sim \chi^2(n-1)$ and 
a two-sided test works. This test is designed against gamma alternatives.
See K.C. Kapur and L.R. Lamberson Reliability in engineering design. Wiley
1977.
A: To test for normality, Anderson-Darling and Shapiro-Wilk are considered the best. For exponential distributions use the Lillerfors test, which is designed specifically for that purpose.
A: Have you considered graphical methods to see how the data behaves? 
Probability graph techniques usually involves ranking the data,
applying the inverse CDF and then plotting the results on the Cartesian plane. This allows you to see if several values deviate from the hypothesized distribution
and possibly account for the reason for the deviation.
A: It seems that you're trying to decide whether to model your data using the normal or the exponential distribution. This seems somewhat strange to me, as these distributions are very different from each other.
The normal distribution is symmetric whereas the exponential distribution is heavily skewed to the right, with no negative values. Typically a sample from the exponential distribution will contain many observations relatively close to $0$ and a few obervations that deviate far to the right from $0$. This difference is often easy to see graphically.
Here is an example where I've simulated $n=100$ observations from a normal distribution with mean $2$ and variance $4$ and an exponential distribution with mean $2$ and variance $4$:

The symmetry of the normal distribution and the skewness of the exponential can be seen using histograms, boxplots and scatterplots, as illustrated in the figure above.
Another very useful tool is a Q-Q-plot. In the example below, the points should approximately follow the line if the sample comes from a normal distribution. As you can see, this is the case for the normal data, but not for the exponential data.

If graphical examination for some reason isn't enough for you, you can still use a test to determine whether your distribution is normal or exponential. Since the normal distribution is a scale and location family, you'll want to use a test that is invariant under changes in scale and location (i.e. the result of the test should not change if you change your measurements from inches to centimetres or add $+1$ to all your observations).
When the null hypothesis is that the distribution is normal and the alternative hypothesis is that it is exponential, the most powerful location and scale invariant test is given by the statistic $$T_{E,N}=\frac{\bar{x}-x_{(1)}}{s}$$
where $\bar{x}$ is the sample mean, $x_{(1)}$ is the smallest observation in the sample and $s$ is the sample standard deviation. Normality is rejected in favour of exponentiality if $T_{E,N}$ is too large.
This test is actually a one-sided version of Grubbs' test for outliers. You'll find this implemented in most statistical software (but make sure that you use the right version - there are several alternative test statistics used for the outlier test!).
Reference for $T_{E,N}$ being the most powerful test: Section 4.2.4 of Testing for Normality by H.C. Thode.
A: As shown in this table, there are a lot of tests for exponentiality. Please refer to this paper for detail (with table above taken from it): Rahman M, Wu H. Tests for exponentiality: A comparative study[J]. Am. J. Appl. Math. Stat., 2017, 5(4): 125-135.
Of course, many of you guys may just want to use a Python package to test whether your data follows a gaussian or exponential distribution. For this end, I suggest you trying Anderson-Darling test, and here is a link scipy.stats.anderson that shows how to use it in Python.
