  The standard model of Z0 is Vω.  The Axiom of Infinity (in one formulation) simply asserts that this set exists.  So, when we add the Axiom of Infinity, the resulting theory Z1 (known as Zermelo set theory with Choice) not only proves the consistency of Z0; it proves that there is a standard model of Z0.  Now the standard model of Z1 is Vω+ω.  The Axiom of Replacement implies that this set exists.  So, when we add the Axiom of Replacement, the resulting theory Z2 (known as ZFC), not only proves the consistency of Z1; it proves that there is a standard model of Z1.