Reid's Undergraduate Algebraic Geometry requires very very little commutative algebra; if I remember correctly, what it assumes is so basic that it is more or less what Eisenbud assumes in his Commutative Algebra!
The trouble with algebraic geometry is that it is, in its modern form, essentially just generalised commutative algebra. Indeed, in a very precise sense, a scheme can be thought of as a generalised local ring. (The structure sheaf $\mathscr{O}_X$ of a scheme $X$ is a local ring object in the sheaf topos $\textrm{Sh}(X)$, and a $\mathscr{O}_X$-module is literally a module over $\mathscr{O}_X$ in the topos.) If you are willing to restrict yourself to smooth complex varieties then it is possible to use mainly complex-analytic methods, but otherwise there has to be some input from commutative algebra.
That said, it is not necessary to learn all of Eisenbud's Commutative Algebra before starting algebraic geometry. Classical algebraic geometry, in the sense of the study of quasi-projective (irreducible) varieties over an algebraically closed field, can be studied without too much background in commutative algebra (especially if you are willing to ignore dimension theory). Reid's Undergraduate Algebraic Geometry, Chapter I of Hartshorne's Algebraic Geometry and Volume I of Shafarevich's Basic Algebraic Geometry all cover material of this kind.
Modern algebraic geometry begins with the study of schemes, and there it is important to have a thorough understanding of localisation, local rings, and modules over them. A scheme is a space which is locally isomorphic to an affine scheme, and an affine scheme is essentially the same thing as a commutative ring. The theory of affine schemes is already very rich – hence the 800 pages in Eisenbud's Commutative Algebra! For general scheme theory, the standard reference is Chapter II of Hartshorne's Algebraic Geometry, but Vakil's online notes are probably much more readable. Volume II of Shafarevich's Basic Algebraic Geometry also discusses some scheme theory. Also worth mentioning is Eisenbud and Harris's Geometry of Schemes, which is a very readable text about the geometric intuition behind the definitions of scheme theory.
Perhaps the most important piece of technology in modern algebraic geometry is sheaf cohomology. For this, some background in homological algebra is required; unfortunately, homological algebra is not quite within the scope of commutative algebra so even Eisenbud treats it very briefly. The first few chapters of Cartan and Eilenberg's Homological Algebra give a good introduction to the general theory but is strictly more than what is needed for the purposes of algebraic geometry. (For example, they allow their rings to be non-commutative.) The last chapters of Lang's Algebra also cover some homological algebra. Chapter III of Hartshorne's Algebraic Geometry is dedicated to the cohomology of coherent sheaves on (noetherian) schemes.
Personally, I think your goal should be to try to get to Ravi Vakil's book Foundations of Algebraic Geometry as quickly as possible. But since he starts with schemes, it is a good idea to get some familiarity with the classical theory of algebraic varieties.
First, you should learn the basic dictionaries ($k$ an algebraically closed field):
\begin{align}
\left\{ \text{regular functions on affine space $\mathbb{A}^n$} \right\} & \longleftrightarrow k[x_1,\dots,x_n]
\end{align}
\begin{align}
\left\{ \text{points of $\mathbb{A}^n$} \right\} & \longleftrightarrow \left\{ \text{maximal ideals in $k[x_1,\dots,x_n]$} \right\}
\end{align}
\begin{align}
\left\{ \text{subvarieties of $\mathbb{A}^n$} \right\} & \longleftrightarrow \left\{ \text{prime ideals in $k[x_1,\dots,x_n]$} \right\}
\end{align}
\begin{align}
\left\{ \text{algebraic subsets of $\mathbb{A}^n$} \right\} & \longleftrightarrow \left\{ \text{radical ideals in $k[x_1,\dots,x_n]$} \right\}
\end{align}
You should also learn the similar dictionary for any affine variety $X$ corresponding to a radical ideal $I$ with coordinate ring $k[X] = k[x_1,\dots,x_n] / I$. As part of this, you'll want to learn about the Zariski topology and you'll need to understand the various forms of Hilbert's Nullstellensatz.
You'll also want to learn what projective space and projective varieties are and learn the analogous dictionaries in that setting. Finally, you'll want to know what a quasi-projective variety is.
You'll need to learn what morphisms (also called regular maps) are in these settings. If you understand the category of quasi-projective varieties (both objects and morphisms), you're off to a good start.
You should also get some familiarity with the function field of an algebraic variety and understand the distinction between rational maps and regular maps, as well as between birational equivalence and isomorphism.
Then it may help to see some basic geometric constructions in a classical setting (Zariski tangent space, singularities, divisors), though you can learn this later if you are willing to accept on faith that quasi-projective varieties (and their generalization to schemes!) are worthwile geometric objects to study even though you don't yet have many tools in your geometric toolbox. Also, the first section of Shafarevich's book has a nice sampling of the types of problems that algebraic geometers are interested in, so it's definitely worth reading, though not necessarily for mastery at this point.
Best Answer
I am not sure you will find a single textbook with all informations. As suggested in comment, one answer could be "look up any book on $D$-modules" (for example these lectures by Bernstein are very good).
Here are some other pointers that could be useful :
These lectures by Ginzburg on non-commutative geometry, that contains a lot of important constructions. I really learned a lot just by reading the first chapters.
These lectures by Gaitsgory on geometric representation theory. It includes the study of category $\mathcal O$ for complex semisimple Lie algebras $\mathfrak g$, and the study of $\mathscr D$-modules, an important example of non-commutative algebras appearing in algebraic geometry.
I also think that books on Hochschild cohomology are relevant : these lectures by Pieter Belmans are geometry-oriented, and this book by Sarah Witherspoon is more algebraic but still could be a good reference.