Super basic questions on statistical process
Answer to your question related to Gaussian Random Process :


  For a Gaussian Random Process, Wide-Sense Stationarity implies Strict-Sense Stationarity as well.


That means, if mean $\mu_X(t)$ is independent of $t$ and is a constant and Auto-correlation function $R_{X}(t, t+\tau)$ is dependent only on time-shift $\tau$, then the Gaussian Process is actually Strict-Sense Stationary.

Why?
Because a Gaussian Random Process, $X(t)$, is defined as that random process for which if we freeze time at $n$ instances to get random variables $\{X(t_1), X(t_2), ..., X(t_n)\}$, then these random variables are jointly Gaussian Distributed. So, we can say that if we have a Gaussian Random Process $X(t)$ and we freeze time at $n$ instances, $t_1, t_2, ..., t_{n}$, then we get a Jointly Gaussian Random vector $\mathbf X$ of $n$ dimension. And for a jointly Gaussian Random Vector, its joint Probability Density is fully determined by its Mean vector $\mathbf \mu_{\mathbf X}$ and Covariance Matrix $\mathbf C$, given by:

$$f_{\mathbf X}(\mathbf x) = \frac{1}{(2\pi)^{\frac{n}{2}}|C|^{\frac{1}{2}}}\cdot \exp(-\frac{1}{2}(\mathbf x - \mu_{\mathbf X})^TC^{-1}(\mathbf x - \mu_{\mathbf X}))$$ 

Also, note that the Auto-Correlation and Covariance Matrix $\mathbf C$ are related for a Wide-Sense Stationary Process, that, all the elements $\mathbf C_{i,j}$ can be computed from the Auto-Correlation function and constant mean of the WSS Random Process. $$\mathbf C_{i,j} = \mathbf E[(X(t_i) - \mu_X)(X(t_j)-\mu_X)] = \mathbf R_X(t_i-t_j) - \mu_X^2$$
Thus, if the mean of a Gaussian Random process is constant $\mu_{X}$ and Auto-Correlation is only dependent on time-shift $\tau$, then the Gaussian Random Process is Strict Sense Stationary, because we know all the joint Distributions completely for any $n$.This is not true for any other Random Process in general.

For detailed definition of Strict-Sense Stationary Random processes and Wide-Sense Stationary Random Process, see the rest of the explanation.

The way I visualize Random processes $X(t, \zeta)$ is: for every outcome $\zeta$ in a random experiment, the outcome $\zeta$ is mapped to a function $x(t, \zeta)$. This is like Random Variable concept. A random variable $X$ is a rule for assigning a value or set of values $x(\zeta)$ to each outcome $\zeta$ of a random experiment. Similarly, a Random process $X(t)$ is a rule of assigning a function $x(t,\zeta)$ to each outcome $\zeta$ of a random experiment. Three important intuitions are as follows:


If we freeze the outcome $\zeta$ of that random experiment, we will get a deterministic function of $t$, $x(t)$. 
If we freeze $t$, then we get a Random Variable $X(\zeta)$.
If we freeze both $t$ and $\zeta$, we get a number $x$.
$$X(t, \zeta) : ensemble \ of \ function \ of \ t \ which \ are \ uncertain$$


Consider a Random process $X(t)$ (dropped $\zeta$ to de-clutter) $\forall \ t \in \mathbb R$. 

$X(t)$ will be called Strict-Sense Stationary(SSS) Process if its statistical properties are time-shift invariant. This means that the random processes $X(t)$ and $X(t+\tau)$ have the same statistics for all $\tau \in \mathbb R$.

What I mean by statistical properties is: If we freeze time at $t$, and get random variable $x(t)$ from the random process $X(t)$, then we can define $f(x(t))$ as the probability density function of $x(t)$. Similarly, we can define joint probability density function of $n$ such random variables $x(t_1), x(t_2),..., x(t_n)$ by fixing time at $t_1, t_2,...,t_n$, as $f(x(t_1), x(t_2),...,x(t_n))$. Then, Strict-Sense Stationary says that:
$$f(x(t_1), x(t_2),...,x(t_n)) = f(x(t_1+\tau), x(t_2+\tau),...,x(t_n+\tau)), \ \forall \ \tau \in \mathbb R$$

We can put $n=1$ in the above expression to figure that, $1^{st}$ order probability density of SSS Random processes are independent of time $t$:
$$f(x(t)) = f(x(t+\tau)) = f(x)$$

Similarly, if we put $n=2$, and look at the $2^{nd}$ order probability density $f(x(t_1), x(t_2))$ will also depend only on time-shift $\tau$ between $t_2$ and $t_1$. Since we can choose any value of $\tau$, consider the case that we choose $\tau = -t_2$ always, then:
$$f(x(t_1), x(t_2)) = f(x(t_1-t_2), x(0))$$
The above expression says that joint density function of $2$ random variables $x(t_1)$ and $x(t_2)$ which we got by freezing time at $t_1$ and $t_2$, is always equal to the joint density function of $2$ random variables $x(0)$ and $x(t_1-t_2)$ which we get when we freeze time at $0$ and $(t_1-t_2)$.
This means that $2^{nd}$ order density function is independent of time and only dependent on the time shift $(t_1-t_2)$.

Wide-Sense Stationary(WSS) Random Processes are those Random Processes which satify following $2$ requirements:


Mean of the Random Process, $\mathbf E[X(t)]$, is independent of time and is a constant.
$$\mathbf E[X(t)] = \mu_X = c$$
Auto-correlation function of the Random Process, $\mathbf R_X$, is only dependent on time-difference $\tau = t_1 - t_2$ 
$$\mathbf E[x(t_1)x^*(t_2)] = \mathbf R_X(\tau)$$


For a WSS Process $X(t)$, the above requirement implies that the variance of the Random Process, $\mathbf E[(X(t)-c)^2]$, is also a constant.
$$\mathbf E[(X(t)-c)^2] = \mathbf E[(X(t))^2] - c^2 = \mathbf R_X(0) - c^2$$

Note that there is no requirement on statistical properties of Wide Sense Stationary Random Processes.

The field of Stochastic Processes deals with these topics and a good starting reference would be Chapter: 9 of "Probability, Random Variables and Stochastic Processes, by A. Papoulis &amp; S. Unnikrishna Pillai"