One thing you could do is go through some mathematical arguments that are already known to you (preferably concerning sets and functions), and convince yourself that they can be formalized using the axioms you're given.
(Later: since functions and relations are only a later chapter, it is not clear to me that you're even supposed to be able to do anything interesting by yourself yet. That is some strange book you have gotten your hands on.)
It is possible that some of them can't, since you seem to be missing the Axiom of Replacement. However it is not needed for much of elementary mathematics.
Google found me a PDF of the book you refer to. It looks like you're also missing the Axiom of Extensionality (which the authors explicitly omit with a justification that makes no sense to me at all). You're also missing the Axiom of Foundation, which is not much of a problem because it has few real applications outside higher set theory.
In general, the entire idea of presenting an axiomatic set theory to a readership that the authors think will be helped by lots of cutesy drawings of bags-within-bags is either extraordinarily bold or raving mad. I tend towards "raving mad", especially because the chapter on "logic" that precedes it only treats propositional logic and doesn't even mention quantification.