Actually proving a new metric-like space to be metrizable is an interesting study in fixed point theory. Though I don't have so much knowledge or idea about all the metrization theorems except for Urishon metrization theorem. So I want to know more about this area. Thank you.
Urysohn showed that every second-countable topological space satisfying T4 is metrizable. The form most recognize as Urysohn's metrication theorem, every second-countable topological space satisfying T3 is metrizable was proved by Tychonoff. Urysohn's theorem can be restated as: A topological space is separable and metrizable if and only if it is metrizable.
There are a lot of examples of topological spaces that cannot be metrizable. The Zariski topology on an algebraic variety is non-metrizable as is the spectrum of a ring - both important in algebraic geometry. The topological vector space of all functions on the real line with point wise convergence is non-metrizable.
Separable metrizable spaces can be characterized as those spaces which are homeomorphic to subspaces of the Hilbert cube.
https://en.wikipedia.org/wiki/Hilbert_cube
To the best of my knowledge, there is not such a book. Maybe professor James F Peters would know better.
Although a classical result, Urysohn theorem only provides a sufficent condition for a space to be metrizable. Nagata-Smirnov theorem gives both sufficient and necessary conditions.
Another common way is to show that a uniformity generating a given topology admits a countable base, see e.g.
Preprint On $\star$-metric spaces
Also, we have the so called Nemytzki-Wilson metrization criterion, which is very useful for spaces endowed with a distance function, see e.g.Article Some remarks on the metrizability of some metric-like structures
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