What is a good introduction to proper forcing? I am aware of Shelah Proper and Improper Forcing, but I heard this book may be somewhat challenging to read. There is also Devlin's The Yorkshiremen's Guide to Proper Forcing and Baumsgartner's Applications of the Proper Forcing Axiom.
Are any of the above good introductions to proper forcing? Are there other surveys in proper forcing that I did not list above that may work better as an introduction?
Thanks.
On
In addition to Asaf's recommendations, you should also check Martin Goldstern's "Tools for your forcing construction": http://info.tuwien.ac.at/goldstern/papers/tools.ps
Once you're familiar with the basics of proper forcing, I recommend checking the wonderful book of Bartoszynski and Judah for some applications in the set theory of the reals (in particular, check chapter 6 for some nice preservation theorems).
Here is a good stretch for understanding proper forcing a little bit.
Start with Jech Multiple Forcing, he covers the basics there. Then re-read the same material in Abraham's Handook of Set Theory chapter, aptly named Proper Forcing.
Being his M.Sc. student, I had the pleasure of having him explain some things in person. So I can only recommend that you buy a plane ticket to Israel and meet with him. But since it seems a bit too much to suggest, here is some insight that I took from him.
The definition of proper forcing seems very unnatural at first (which is why I suggested Jech, which uses a much simpler definition), but after doing a few examples - in particular Baumgartner's forcing which adds a club with finite conditions - the definition using countable submodels and master conditions becomes much clearer.
Which is why I suggested above to begin with Jech, then re-learn most of the things there from Abraham's chapter (after having a rudimentary understanding of a definition of proper forcing it becomes easier).