I want to offer a possibly heretical opinion based on conversations I've had with people who do algebraic geometry, especially Joe Harris. I think that it is not necessary to know very much commutative algebra in order to study and understand algebraic geometry. The fundamental objects algebraic geometry studies are very concrete and intuitive, and even when you encounter non-reduced schemes, or jump into positive characteristic to study number theory using AG tools, there are a few standard techniques of translating results from familiar settings into new ones, and much of the formalism surrounding them can be treated as a black box. If you need to understand some exercise in Hartshorne, for example, then as long as you want to know the geometry, you can safely skip and assume the commutative algebra exercises, or make additional assumptions about your rings that make things easier technically.

There are two situations when it's good to know what's under the hood of a car: one is when it breaks, the other is when you need to make a new car. Neither of them should prevent you from getting behind the wheel and learning to drive. And if you know what a car does and when it breaks, you'll be surprised by how much more sense the fancy technology under the hood will make.

That said, a lot of great math has been done by digging around under the hood. If you want a recommendation book-wise, I'd go for Atiyah-MacDonald.