What is the expectation of a random variable raised to the $n$th power? If $Y=X^n$, with $n$ and the expectation and variance of $X$ known, what is the expectation and variance of $Y$?
To give this an answer, as Stefan Hansen comments:


  Knowing the expectation and variance of X is not enough information to know the moments of all orders.


As an explicit example, let $X$ be normally distributed with mean zero and variance 1, and let $W$ be +1 or -1 with probability 1/2 (the Rademacher distribution).  Then $EX = EW = 0$ and $E X^2 = E W^2 = 1$, but $E X^4 = 3$ while $E W^4 = 1$.

In the special case of the normal distribution, the distribution of a normal random variable are completely determined by its mean and variance, hence so are its higher moments.  You can find various formulas for those moments on Wikipedia.
Here's a partial answer: If $X$ is a Bernoulli variable, and $n\gt0$, then $E[X^n]=E[X]$ and $Var(X^n)=Var(X)$.