I manage the New Foundations Home Page , where you could look for enlightenment about this subject if I had updated it anytime recently... I am planning to work on this.
I have written a book, Elementary Set Theory with a Universal Set , which HAS APPEARED as no. 10 in the Cahiers Series of the Center for Logic in the department of philosophy at the Catholic University of Louvain (but the book is now out of print: see the next paragraph). See here for a local file with order information, or look at the Cahiers site. This book is roughly parallel to Halmos's Naive Set Theory but the development is carried out in NFU (New Foundations with urelements) instead of ZFC. Earlier electronic versions of the book contain various inaccuracies and infelicities; please consider getting the final version! There are of course some residual inaccuracies and infelicities, here humbly noted in an errata list (text) or errata list (PDF).
With the permission of the publishers, the book is now posted online here. This is an ad hoc corrected version; there will be an official second edition online eventually.