How to approach proofs similar to "Show a group, $G$, is infinite if $G = \langle r, s, t\mid rst = 1\rangle $"

$G$ is the set of words on $r,s,t$ subject to the relation $rst=1$.

The relation $rst=1$ means that you can replace every occurrence of $t$ by $(rs)^{-1}=s^{-1}r^{-1}$.

Therefore, $G$ is the set of words on $r,s$, that is, the free group in two letters.

Alternatively, the set $\{1,r,r^2, r^3, \dots \}$ is an infinite subset of $G$ because these words do not contain $s$ or $t$ and so cannot be further reduced or to one another.

(By words on $S$, I mean words on the elements of $S$ and their inverses.)

One thing I often find clarifying is to try adding relations. If you still get an infinite group after you added a relation then you must have started with an infinite group.

Here, for instance, set $r=e$. Then the new group is generated by $s,t$ with $s=t^{-1}$. Hence it is generated by $t$ with no relations, so the new group is $\mathbb Z$. As that is infinite, so must $G$ have been.

Consider $f:\{r,s,t\}\rightarrow\mathbb{Z}$ defined by $f(r)=1, f(s)=-1, f(t)=0$, $f(r)+f(s)+f(t)=0$ implies that $f$ extends to a morphism of groups $g:G\rightarrow\mathbb{Z}$. The fact $g(r^n)=n$ implies that$g$ is surjective and $G$ infinite.