What is the Galois group of $t^4+2$ over $\mathbb{Z}_5$ I see that $t^4+2$ is irreducible. I now want to find an extension that contains the roots of this - but I'm not sure how to do that. 

You are asking for the roots of $x^4=-2=3$ over $\Bbb F_5$, and since $3$ is a primitive fourth root of unity in $\Bbb F_5$, you are asking for the smallest field that contains the sixteenth roots of unity.
In other words, you’re looking for the smallest field $\Bbb F_{5^n}$ with $16|(5^n-1)$, thus $16k=5^n-1$, thus $5^n\equiv1\pmod{16}$. The powers of $5$ are $5$, $5^2\equiv9$, $5^3\equiv-3$, $5^4\equiv-15\equiv1$.
So $\Bbb F_{5^4}=\Bbb F_{625}$ is your splitting field, of degree four over the prime field, and so your Galois group is cyclic of order four.
