I know that $Q$ is a field, which makes $Q[x]$ a PID, which means there are none.
I'm having trouble with the notation for ideal generators, and i know the $Z[x]$ has to do with something that looks like $(2,x)$. Could someone explain this notation to me and also the general outlines of solving these problems?
The example $(2,X)$ is a good one. It means the ideal generated by $2$ and $X$. You can think of this vaguely like a subspace of a vector space.
Explicitly $(2,X) = \{ 2 p + X q \colon p, q \in \mathbb{Z}[X]\}$.
By construction it is an ideal. You need to show it is not generated by a single element. Assume a generator $g$. Then $g \mid 2$ and $g \mid X$. However, the only elements that do this are $1$ and $-1$. So the ideal generated by $g$ is the full ring, and thus $(2,X)$ would be the full ring. This is however not the case. (Prove it.)
For $\mathbb{Q}[X,Y]$, consider $(X,Y)$ and do roughly the same thing.
Both examples fall under the strategy: pick an element $f$ that has no nontrivial divisors and try to find another element $h$ that has no nontrivial divisors either and such that $(f,h)$ is not the full ring.