Method of finding a p-adic expansion to a rational number Could someone go though the method of finding a p-adic expansion of say $-\frac{1}{6}$ in $\mathbb{Z}_7?$ 

The short answer is, long division.
Say you want to find the $5$-adic expansion of $1/17$. You start by writing
$$
\frac{1}{17}=k+5q
$$
with $k \in \{0,1,2,3,4\}$ and $q$ a $5$-adic integer (that is, a rational number with no powers of $5$ in the denominator). Then $k$ is the first term in the expansion, and you repeat the process with $q$ to find the remaining terms.
In this case, we have
$$
\frac{1}{17}=3+5\left(-\frac{10}{17}\right)
$$
and so the first term is a $3$. Continuing similarly,
$$
-\frac{10}{17}=5\left(-\frac{2}{17}\right)
$$
and so the second term is a $0$,
$$
-\frac{2}{17}=4 + 5\left(-\frac{14}{17}\right)
$$
and so the third term is a $4$,
$$
-\frac{14}{17}= 3 + 5\left(-\frac{13}{17}\right)
$$
and so the fourth term is a $3$, and so on.
Eventually, you'll hit a remainder for the second time, and then the expansion will start repeating (just as when you're computing the repeating decimal expansion of a fraction using ordinary long division).
