In my site http://settheory.net I provide explanations on the main issues and paradoxes at the foundations of mathematics, with the articulation between set theory and model theory. I explain in details the difference of meaning between sets and classes, and the relative degree of justifications of different axioms.
These explanations clearly show in particular that there is no universal set in nature, so that axiom systems admitting one such as New Foundations, may be studied as logical curiosities but cannot be accepted as a "natural" foundation for mathematics.
I give a justification for the consistency of ZF (showing that the one doubtful axiom is the powerset axiom, which we need anyway as far as I know), and find it relevant as a basis for the work of professional set theorists studying relative consistency issues.
But I propose another formalization accepting functions as fundamental objects aside sets, and many symbols instead of one, that I consider more appropriate to start mathematics from scratch : to facilitate the understanding of basic mathematical concepts, make the first developments of set theory simpler and more intuitive, and better fit with the common practice of mathematics that uses many symbols. Indeed I see the usual construction of ordinary mathematical tools from the mere membership predicate as unnatural, overcomplicated and irrelevant for beginners.