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.
Robert E. Mosher and Martin C. Tangora, Cohomology Operations and Applications in Homotopy Theory, Harper and Row (1968)
Marcelo Aguilar, Samuel Gitler, Carlos Prieto, Algebraic topology from a homotopical viewpoint, Springer (2002)