Stanley Farlow's "Partial Differential Equations for Scientists and Engineers" would be my suggestion for an introduction. It's a Dover book, so it's cheap. Look at the book as a set of simple introductions to a large number of PDE related topics, many of which have applications beyond PDEs, e.g., the Laplace transform, conformal mapping, perturbation methods, etc. The book would disappoint anyone looking for rigor. You can refer to a more advanced book for that.<p>The book has relatively few exercises, but the ones present are definitely worth your time. I recall working out a large fraction of them before I took a PDE class as an undergraduate. Dover will also send you a scan of the solutions manual if you contact them. As I recall a few exercises had mistakes in their solutions, but I had no trouble spotting the mistakes as a novice.
Consider a book on numerical methods first perhaps, and then something more theoretical to dive into the theory, I went to Berkeley so Evans was a popular book in that regard.
We used Haberman. I think you can find a solution book. I'm not really sure what constitutes a gentle introduction. Half the class probably flunked or should have!