What is a good PDE book suitable for self study? I'm looking for a book that doesn't require much prerequisite knowledge beyond undergraduate-level analysis. My goal is to understand basic solutions techniques as well as some basic theory.
I would recommend:
Fritz John, Partial Differential Equations (Applied Mathematical Sciences) ISBN: 0387906096. It is a classical Springer book that contains what you ask for.
Google Books might be a good start before you make your final decision.
Evans' book  is used in many curricula and is quite famous. It is easy to read for people with background in mathematics and analysis, it has many examples and exercises, and it covers quite diverse PDE topics. However, there are many alternatives if this book is not well-suited to the reader's background or objectives. Among others, one may prefer
the book by Strauss  -- one of my favorites -- which is very easy to read.
the book by Olver  -- one of my favorites -- is very easy to start with, since designed for undergraduate level.
the book by Arnold  for people interested in geometrical explanations, but not as easy for a first reading as the two previous books.
the book by Courant and Hilbert  is a classical reference in the spirit of mathematical physics, but it may not fit well for everybody.
 L.C. Evans, Partial Differential Equations, 2nd ed., American Mathematical Soc., 2010.
 R. Courant, D. Hilbert, Methods of Mathematical Physics Vol. 2: Partial Differential Equations, Wiley-VCH, 1962.
I like Karl E. Gustafson's Introduction to Partial Differential Equations and Hilbert Space Methods. It was certainly readable after an advanced calc sequence. You will find a few short and worthwhile conversational paragraphs throughout the book. He also uses the technique of revisiting interesting concepts from different perspectives throughout the book. And it's a Dover paperback, so it's cheap.
Logan's Applied Partial Differential Equations might be suitable for you if you want a (relatively) quick overview of the subject, since it's not very long (about 200 pages). It's aimed at undergraduates in math, engineering and the sciences.
My favorite undergraduate texts on PDEs are the older ones-I don't like more recent undergraduate texts because they either require too much prerequisites or they're just not very well organized. An undergraduate text on PDE's is really a course on the classical theory that doesn't use graduate level analysis-when you look at it from that point of view, it makes sense to use the older books for a first course.
The book I first learned PDEs from was Elementary Partial Differential Equations by Paul W. Berg and James. L. McGregor. It is extremely clear, very gentle and covers all the basic with just a background in calculus needed. It also has many wonderful problems. Somewhat more sophisticated but equally good is Introduction to Partial Differential Equations with Applications by E. C. Zachmanoglou and Dale W. Thoe.It's a bit more rigorous, but it covers a great deal more, including the geometry of PDE's in R^3 and many of the basic equations of mathematical physics. It requires a bit more in the way of prerequisites: some advanced calculus of functions of several variables, some linear algebra and basic differential equations. But it's beautifully written and covers a lot more-and it's available in Dover paperback. If I had a gun to my head and could only use one book, that's the one I'd use.
I haven't read a lot of PDE material, but I enjoyed Taylor's book. It's quite well-written, and also contains introductory material (like Lie derivatives), since it does things on manifolds.
I would also like to add Peter Olver's "Introduction to partial differential equations" to this growing list. It was published in 2014 and is very suitable for self-studying. The preface does a good job of explaining the various topics and prerequisites required to understand PDEs which I hadn't appreciated until I read it in the book-highly recommended. Solutions to about 20% of exercises are available on the author's website.
I think you cannot get anything better than Evans' book. Its size may be a little scaring, but it is the most clear and well written book on the subject I ever met.