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Is there a good (free) online resource for learning axiomatic set theory? I am no mathematician, but I have basic familiarity with calculus, combinatorics, and basic computability theory. Wikipedia, in typical encyclopaedia fashion, is a bit too dense, and lacks exercises.

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    what is wrong with reading a good old fashioned book ? – 2017-01-20
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    You can start with Stephen Simpson, [Foundations of Mathematics](http://www.personal.psu.edu/t20/notes/fom.pdf) (2009). – 2017-01-20
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    "An old-fashioned book" is generally more difficult to access. – 2017-01-20
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    Almost all math books can be found online these days. – 2017-01-20
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    @MauroALLEGRANZA thank you! Although it seems most axiomatic set theory starts a bit late in the book, and runs for only a few dozen pages before the book is over :) But it definitely seems a good starting point. Anything AFTER that? – 2017-01-20
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    @ReneSchipperus then great, a free online math book is a free online resource and qualifies as an answer. But what's wrong with being open about *other* resources too? – 2017-01-20
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    Nothing is wrong, its just that I got the impression from your question that you wanted a website. For those people serious about learning set theory the books Kunnen, Set theory, and Jech, Set Theory remain the two basic resources. I would recommend starting with Kunnen. Of course there are other books but eventually one ends up looking at those two. – 2017-01-20
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    @ReneSchipperius I ... *found* a copy of the Kunen (Kunen, right? Not KunNen) online :) It's from 1980. Isn't it a bit old? But delving into it it seems to have far more substance than any other suggestion I received so far. I would accept it as an answer! – 2017-01-20
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    I was once in the library and there were two students looking for a calculus text. One student found a book and his friend looked at the date and said "1989 that is a bit old isnt it ?" and the student replied "Dude, calculus was invented in then 18 th century." – 2017-01-20

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I think the best book is "Axiomatic Set theory" by Patrick Suppes.

Also note this PDF is quite good (by Robert Andre). Hope it helps.

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    Oh thanks. It looks quite well done. I would upvote your answer, but it seems I have not enough reputation yet (11/15 needed). – 2017-01-20
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    I just gave you the rep. you need. – 2017-01-20
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    @ReneSchipperus thanks :) – 2017-01-20
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    @blackbadger Also Kunen's book on set theory is very good ( just read the comments!). Happy reading! – 2017-01-20
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There is a preliminary version of the introductory text on Set Theory by W. Weiss in his website. It is a very nice material.

Once you cope with the basic stuff and after you've seen some material on Martin's Axiom, you could check out the notes of Forcing by Itay Neeman. There he refers to some notes by J. Palumbo, but I think those are not available online any more. There were pretty useful for me.

Concerning book suggestions, I completely agree with @ReneSchipperus. Now, there is a 2011 Set Theory by Kunen. It's a slightly different book on exactly the same subject; perhaps more streamlined. Both the old and new are excellent choices, along with Jech's 3rd edition. For more elementary suggestions, see this answer.