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.