This isn't a fun answer, but you basically can't without substantial background in undergrad mathematics. The subject has some of the highest prerequisites of any upper undergrad/lower grad level math course, as it touches on (and uses) just about everything you'd learn through an undergrad math degree.

I don't want to discourage you, I'm just being realistic. If you want to work towards algebraic geometry, you can certainly do that. You'll need to first master linear algebra and abstract algebra. You should have a strong understanding of fields, groups, rings, vector spaces and modules. Someone else mentioned commutative algebra - that is more of a circular dependency with algebraic geometry than a hard one. It's good to have walking in, but realistically you can't master the subject without knowing algebraic geometry.

You'll also need analysis, in particular complex analysis for curves. Real analysis and topology should also be covered but I suppose with tenacity you could get by without them.

To translate these into concrete suggestions, in your position I'd try to work through the following, in order:

1. Linear Algebra Done Right (Axler)

2. Abstract Algebra (Dummit & Foot)

3. Complex Analysis (Ahlfors)

4. Algebraic Curves (Fulton)

The last one is a standard upper undergraduate introduction to the subject.

If possible you should organize a study group or take a class though, because trying to learn math on your own from a textbook is rough.

The difficulty with learning 'modern' algebraic geometry is not only is it very dense and general, but that means the original motivation can become lost.

So I think understanding Weil conjectures are key for modern algebraic geometry. And it's always easier to understand algebraic curves (algebraic geometry with dimension 1) and their connection to Riemann surfaces (algebraic curves over the complex numbers with analytic rather then algebraic structure), as they provide motivation for many of the results and constructions.

A good introduction to Algebraic Curves and the Weil conjectures I've found is following

https://math.mit.edu/~poonen/papers/curves.pdf

For general algebraic geometry, JS Milne's notes are rather good

https://www.jmilne.org/math/CourseNotes/ag.html

and for an introduction to commutative algebra Atiyah-Macdonald's book is great.

My favorite intros to algebraic geometry are these:

Igor R. Shafarevich, Basic Algebraic Geometry, two volumes, third edition, Springer, 2013.

Phillip Griffiths and Joseph Harris, Principles of Algebraic Geometry, 1994. (Especially nice if you like complex analysis, differential geometry and de Rham theory.)

David Eisenbud and Joseph Harris, The Geometry of Schemes, Springer, 2010. Currently at https://www.maths.ed.ac.uk/~v1ranick/papers/eisenbudharris.p...

But whenever I want to learn some new field of math I start with Wikipedia. This is a pretty good overview:

https://en.wikipedia.org/wiki/Algebraic_geometry

Depends how much math you know, but Chapter 15 (Commutative Rings and Algebraic Geometry) of Dummit and Foote may be a good starting place.

I highly recommend the book "An Invitation to Algebraic Geometry" by Karen E. Smith et. al., as an introduction. It'll give you a taste of the field, without delving too deeply into technicalities.

You'll need a good background in commutative algebra to learn algebraic geometry, if you have that and want to see the modern approach, schemes and everything else that Grothendieck did, I suggest Vakil's notes, they're freely available from his homepage. (Disclaimer: I only studied the first 15 chapters and the one on Kähler differentials which should be the 21st, but I suppose the second half is as good as the first, I'll find out for sure next term)

If you do have some math background, "An Invitation to Algebraic Geometry" by Karen Smith et al is a lovely, slim volume that gets to important ideas right away.

"Ideals, Varieties, and Algorithms" is an "invitation to computational geometry" by Cox, Little, and O'Shea. I think parts of it might really appeal to HN readers and it's supposed to be for an undergrad math major audience.

Atiyah-Macdonald is a small but dense book with most of the commutative algebra you'll need to start learning algebraic geometry. You'll need to know some abstract algebra as a prerequisite (groups, rings, fields; covered in undergraduate algebra courses).

Try some projective geometry to get your feet wet.