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I would like to know if there is a book on Geometry from a modern point of view which can guide me through the basics of the subject into more advanced graduate level material. I’m looking for a book on geometry that as closely resembles the books on Analysis by Herbert Amann and Joachim Escher in structure and presentation as possible because I really like those texts.

Do tell me about any other valuable sources that can help me with this

Thank you :)

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I'm not familiar with Amann and Escher's book, but the classic introductory text to modern (algebraic) geometry is Hartshorne's book "Algebraic Geometry". This requires a good background in commutative algebra and basic topology, but it is worthwhile. Other texts that come to mind would be Eisenbud and Harris' "The Geometry of Schemes", Shafarevich's "Basic Algebraic Geometry", or Mumford's "Red Book of Varieties and Schemes". Also, though I'm not as familiar with it, I have heard that Ravi Vakil's book "The Rising Sea: Foundations of Algebraic Geometry" (available for download online) is very good.

If you're interested more in differential geometry, perhaps Lee's "Introduction to Smooth Manifolds" might be a good place to start, but maybe others have better suggestions in this respect.

A more comprehensive list of texts can be found in the responses here.

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