I want to follow a Ph.D. programme on geometric analysis. The main focus are Monge-Ampere equations and convex level set of $p$-harmonic equations. The theory of PDEs is very familar to me. However, I have little knowledge of the geometry. Therefore, I want some readable texts on this issue. Since (linear) functional analysis and nonlinear analysis are my stong points, I want the texts to describe the geometry in terms of analysis.
Thierry Aubin, A Course in Differential Geometry would be suitable for your goals. It is self-contained (includes the necessary background from differential topology and reaches the Riemannian geometry), and features a chapter with an introduction to Yamabe problem (one of the central topics in the geometric PDEs). A delicious part of this book is a lot of solved exercises that make your acquaintance with the subject much faster.