Textbook for Partial Differential Equations with a viewpoint towards Geometry.

The standard PDE textbooks would serve you well as a start. For example Fritz John's Partial Differential Equations and LC Evans' Partial Differential Equations are both very good for just the analysis aspects.

In parallel you can consult Michael Taylor's 3 volume set titled, again, Partial Differential Equations. Taylor's books develops similar breadth and slightly more depth as Evans' book, but also with an eye toward geometry.

Once you get a few things under your belt, J Jost's Riemannian geometry and geometric analysis is a classic in the field.

For more advanced topics, particular related to geometry, two very good books have been written by Thierry Aubin: Nonlinear Analysis on Manifolds, Monge-Ampere Equations and Some Nonlinear Problems in Riemannian Geometry.

If you particularly want to study elliptic type problems, you can also consider going from John's book to Fang-Hua Lin's Elliptic partial differential equations then to Gilbarg & Trudinger on the one hand, and Caffarelli & Cabré on the other for the analysis.