For example, how can I show that $\mathbb{Q}$ is the fraction field of $\mathbb{Z}$? Or that $\mathbb{C}$ is the fraction field of $\mathbb{R}$?
I understand that $\mathbb{Z}$ is a subring of $\mathbb{Q}$ & each r in $\mathbb{Q}$ can be written as a fraction r = a/b with a,b in $\mathbb{Z}$ and no proper subfield of $\mathbb{Q}$ has that property. But is there some general way to show this for the other number systems?
The universal mapping property of localizations (e.g. fraction fields) yields an easy test for (unique) isomorphism, cf. $3.2$ below, from Atiyah & MacDonald, Commutative Algebra, p. $39$.
In OP: $\:\!A\,$ is a domain and $\,S\,$ is the set of nonzero elements in $A,\,$ so $\,S\,$ contains no zero-divisors, so condition $(ii)$ in $3.2$ simplifies to $\,g:A\to B\,$ is an injection, so $3.2$ specializes to
Corollary $\,B\,$ is isomorphic to the quotient field of $A\,$ if $\,B\,$ contains an isomorphic image $\bar A$ of $A$ such that every $\,0\neq a\in A\,$ maps to a unit $\, \bar a = g(a)\,$ in $B,\,$ and every $\,q\in B\,$ is a fraction over $\bar A,\,$ i.e. $\,q = \bar a_1 \bar a_2^{-1}\,$ for some $\,a_i\in A,\, a_2\neq 0$.
Remark $ $ More generally an analogous corollary holds for the total ring of fractions of a commutative ring - which inverts every $\,s\in S = $ all regular elements (non-zero-divisors).