Good 1st PDE book for self study

The book by Strauss is pretty good for a first course. For a second one the book by Evans is nice but it requires some knowledge of measure theory and functional analysis.

Partial Differential Equations for Scientists and Engineers by Farlow. It's Dover, so it's cheap. And it's a great first intro - very applied. If you want to follow on with a more rigorous one, you can't beat Evans (Springer - ISBN13: 978-0821207729)

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.

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.