I've been wanting to start learning about some algebraic geometry and after talking to some grad students of my institute I've noticed that the overall recommendation is to start by learning some of the classical way to do it, meaning without the theory of schemes. The problem is 99% of the department learnt this way by the hands of a professor that is already retired and so I would like to know some recommendations of good texts about the subject.
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2This kind of question has been asked many, many times before. See, for instance here, here, and here just to name a few. In particular, the last of these links has a huge list which you'll likely find helpful. I'll also suggest the more recent book algebraic geometry: a problem solving approach which encourages you to build much of the theory yourself, through a series of (quite well guided) exercises. – HallaSurvivor Aug 27 '23 at 22:43
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You could look at my lecture notes https://people.maths.ox.ac.uk/rossler/mypage/pdf-files/CAGnotes.pdf – Damian Rössler Aug 28 '23 at 18:37
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I suggest focusing on a book like Principles of Algebraic Geometry by Griffiths and Harris.
Getting a solid foothold in the context of compact complex surfaces (algebraic curves) and "good old-fashioned" algebraic geometry over the complex numbers goes a long way!
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That said, a newcomer should probably skim (or straight-up skip) Chapter 0 and just refer back for statements as needed. – Tabes Bridges Aug 28 '23 at 04:12