Further reading on the $p$-adic metric and related theory. In his book Introduction to Topology, Bert Mendelson asks to prove that
$$(\Bbb Z,d_p)$$
is a metric space, where $p$ is a fixed prime and 
$$d_p(m,n)=\begin{cases} 0 \;,\text{ if }m=n \cr {p^{-t}}\;,\text{ if } m\neq n\end{cases}$$
where $t$ is the multiplicty with which $p$ divides $m-n$. Now, it is almost trivial to check the first three properties, namely, that
$$d(m,n) \geq 0$$
$$d(m,n) =0 \iff m=n$$
$$d(m,n)=d(n,m)$$
and the only laborious was to check the last property (the triangle inequality). I proceeded as follows:
Let $a,b,c$ be integers, and let
$$a-b=p^s \cdot k$$
$$b-c=p^r \cdot l$$
where $l,k$ aren't divisible by $p$.
Then $$a-c=(a-b)+(b-c)=p^s \cdot k+p^r \cdot l$$
Now we have three cases, $s>r$, $r>s$ and $r=s$. We have respectively:
$$a-c=(a-b)+(b-c)=p^r  \cdot(p^{s-r} \cdot k+  l)=p^r \cdot Q$$
$$a-c=(a-b)+(b-c)=p^{s} \cdot(   k+p^{r-s} \cdot l)=p^s \cdot R$$
$$a-c=(a-b)+(b-c)=p^s \cdot (k+l)=p^s \cdot T$$
In any case,
$$d\left( {a,c} \right) \leqslant d\left( {a,b} \right) + d\left( {b,c} \right)$$
since
$$\eqalign{
  & \frac{1}{{{p^r}}} \leqslant \frac{1}{{{p^s}}} + \frac{1}{{{p^r}}}  \cr 
  & \frac{1}{{{p^s}}} \leqslant \frac{1}{{{p^s}}} + \frac{1}{{{p^r}}}  \cr 
  & \frac{1}{{{p^s}}} \leqslant \frac{1}{{{p^s}}} + \frac{1}{{{p^s}}} \cr} $$
It might also be the case $k+l=p^u$ for some $u$ so that  the last inequality is
$$\frac{1}{{{p^{s + u}}}} \leqslant \frac{1}{{{p^s}}} + \frac{1}{{{p^s}}}$$
$(1)$ Am I missing something in the above? The author asks to prove that in fact, if $t=t_p(m,n)$ is the exponent of $p$, that
$$t\left( {a,c} \right) \geqslant \min \left\{ {t\left( {a,b} \right),t\left( {b,c} \right)} \right\}$$
That seems to follow from the above arguement, since if $s \neq r$ then
$$t\left( {a,c} \right) = t\left( {a,b} \right){\text{ or }}t\left( {a,c} \right) = t\left( {b,c} \right)$$
and if $s=r$ then
$$t\left( {a,c} \right) \geqslant t\left( {a,b} \right){\text{ or }}t\left( {a,c} \right) \geqslant t\left( {b,c} \right)$$
$(2)$ Is there any further reading you can suggest on $p$-adicity?

What you have done seems correct. Also notice that from what you have done you get that
$$
d(a,c) \leq \max \{d(a,b), d(b,c) \},
$$
which is stronger than the triangle inequality.
For an introduction to $p$-adic numbers, I would suggest Fernando Gouvea's $p$-adic Numbers: An introduction. It should be easy for an undergraduate to understand the book, and I think it is a very nice introduction.
Edit: I should add that it's not easy for any undergraduate. One should've had a few courses which have a lot of proofs, and not just the standard calculus courses.
