Equivalently, if the cohomology theory has a classifying space (as it does for all usual notions of cohomology, in particular for all generalized (Eilenberg-Steenrod) cohomology theories) then, by the Yoneda lemma, cohomology operations are in natural bijection with homotopy-classes of morphisms between classifying spaces.
(This statement is made fully explicit for instance below def. 12.3.22 in (Aguilar-Gitler-Prieto).
every universal characteristic class is a cohomology operation.
Steenrod’s original colloquium lectures were published as:
Textbook accounts include the following.
Stanley Kochmann, section 2.5 and 3.5 of Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996
Marcelo Aguilar, Samuel Gitler, Carlos Prieto, Algebraic topology from a homotopical viewpoint, Springer (2002)
A treatment of the differential refinement of cohomology operations is in