A detailed discussion of the early history of set theory is in :
The starting point are Dedekind and Cantor.
Dedekind, in his exposition of the theory of fields (1871) writes [§159] :
By a field [Körper] we mean an infinite system [System] of real or complex numbers [...].
In a draft for Zahlen, written between 1872 and 1878, he introduces the concepts of system and of mapping ["Die Begriffe des Systems, der Abbildung"].
In Was sind und was sollen die Zahlen? (1888) we find also : "zusammengesetze System" and "Gemeinheit der Systeme" ["compounded system" (union - Def.8), and "community of systems" (intersection - Def.17)].
Cantor in the years 1863-64 introduced his theory od point-sets, and in 1872 speaks of set of points [Punktmenge].
Up to now, we have "sets of" something : numbers or points.
The "pure" notion of set has been introduced by Cantor in 1879 :
I say that a manifold (a collection, a set) of elements that belong to any conceptual sphere is well-defined ["Eine Mannichfaltigkeit (ein Inbegriff, eine Menge) von Elementen, die irgend welcher Begriffsphare angehoren"]
followed in 1883 by the "mature" concept :
By a manifold or a set I understand in general every Many that can be thought of as One, i.e., every collection of determinate elements which can be bound up into a whole through a law, and with this I believe to define something that is akin to the Platonic εἶδος or ἰδέα. ["Unter einer Mannichfaltigkeit oder Menge ...].
We can find class into the claculus of class of The Algebra of Logic Tradition : from Boole to Schröder.
Peano in Arithmetices principia (1889, page x) uses classis, followed by Russell (Principles, 1903). Of course, also Frege did [Appendix to vol.2 of Grundgesetze, discussing Russell's paradox, also if class is a "derived" notion for Frege, "concept" [Begriffe] being the primitive one] :
I cannot see how arithmetic could be given a scientific foundation, how numbers could be
conceived as logical objects and introduced, if it is not allowed - at least conditionally - to go from a concept over to its extension. Can I always speak of the extension of a concept, of a class?
Zermelo (1908) starts from Cantor's definition of set (1895) as "a collection, gathered into a whole, of certain well-distinguished objects of our perception or our thought" [Unter einer 'Menge' verstehen wir jede Zusammenfassung ...]
He introduce also the notion of "Klassenaussage" ('class-statement') meaning a logical expression with one variable, what Russell called a propositional function.
Finally, von Neumann (1925) speaks of : set [Menge] and class [Bereich] :
"sets" are the sets that (in the earlier terminology) are "not too big", and "classes" are all totalities [Gesamtheiten] irrespective of their "size".
See also The Early Development of Set Theory.