I know that all finite fields of the same size are isomorphic to one another. I also know that if a polynomial $f(x)$ is irreducible over $\mathbb{Z}[x]$ and of degree $n$ then
$$ \frac{\mathbb{Z}_k[x]}{(f(x))} \cong \mathbb{Z}_{k^n}$$
For instance, we should have that $$ \frac{\mathbb{Z}_5[x]}{(x^2+2)} \cong \mathbb{Z}_{25}$$
However, I'm unsure of how to construct an explicit isomorphism between the two. I want an isomorphism so I can identify things like every generator of the field on the LHS (which will correspond to the inverse under the isomorphism of $2, 3, 4, 6, 7$ etc.). What is an isomorphism for this specific case and how do I then generalise this?
Your method of constructing a field of $25$ elements by the quotient ring of a quadratic polynomial is fine,
but it is a misconception that the ring of integers modulo $25$ is a field.
The ring $\mathbb Z_{25}$ has non-zero zero divisors -- for example, in it $5\times5=0$ even though $5\ne0$ --
so it is not a field. Integers modulo $n$ are a field when $n$ is prime.
The elements of the field of $25$ elements are of the form $a+b\alpha,$
where $a,b\in\mathbb F_5=\mathbb Z_5$ and $\alpha$ is a root of a quadratic polynomial that is irreducible in $\mathbb Z_5$.