Here and here are examples of questions I've answered with little more than the second isomorphism theorem.
The second one comes up a lot in modular arithmetic.
If you working in the integers modulo $16$? Maybe you'll want to reduce modulo $8$. Can you do that? And if so, do you get the integers modulo $8$?
$$ (\mathbb{Z} / 16\mathbb{Z}) / (8\mathbb{Z} / 16\mathbb{Z}) \cong \mathbb{Z} / 8\mathbb{Z} $$
Ah, phew!
The second isomorphism theorem often comes up when you want to do calculations with a quotient ring by lifting the problem to the original ring (e.g. because it is easier to work with, such as the integers or being a polynomial ring)
Now I'm working in the even numbers but I need to also work modulo $3$. How can I make sense of that? I suppose I need to mod out by the intersection of $3\mathbb{Z} \cap 2\mathbb{Z}$. What does that work out to?
$$2\mathbb{Z} / (2\mathbb{Z} \cap 3 \mathbb{Z})
\cong (2 \mathbb{Z} + 3 \mathbb{Z}) / 3 \mathbb{Z} \cong \mathbb{Z} / 3\mathbb{Z}$$
The third isomorphism theorem often comes up when you have several moduli to work with, and want to understand one ideal modulo the other.