I have several works on commutative algebra, including Atiyah-Macdonald, Zariski - Samuel, Eisenbud, Northcott, Matsumura, Milne, and Dieudonne's Topics in local algebra, as well as chapters in general algebra books, such as Lang, Dummitt and Foote, Hungerford, Jacobson, Mike Artin, and Van der Waerden. Out of all these, I find Miles Reid's little book the most useful (although everything actually included in Mike Artin's book is helpful) in the sense of being easy to read, insightful, and limited in its goals. I find I benefit from reading books aimed at people with far less training them myself. I.e. as a postgraduate myself, I often benefit from an explanation that is aimed at graduates or even undergraduates. Atiyah - Macdonald is very authoritative, and the proofs are very efficient and slick and correct, but it is the sort of book whose explanations go "in one ear and out the other" at least for me. The exercises in A-M are also frequently hard for me, whereas the ones in Reid are not only easier, but also more instructive. I did consult A-M for a treatment of general valuation theory, which Miles omits.
To be fair, I think one reason Miles' book is preferable to me, is that he had A-M available for the proofs and only had to augment the insights, improve the readability, and create better exercises. I also have only the earlier work by Matsumura, his Commutative Algebra. His later work Commutative Ring Theory is widely considered to be easier to learn from, and perhaps benefited from its translation by Miles Reid. So while those other books are ones I have spent time in and then stopped, only to return to the same topic later having forgotten it, Miles' book seems to be one that I think I would enjoy reading all of, and then setting it aside, having actually learned it. So far I have read chapters 5,6,7,8, but benefited so much that I actually went back to chapter 1, and learned something. Hence I am tempted to read 2,3,4, even though they had seemed too elementary at first glimpse. I am also inclined to return afterwards to A-M to see if it then is more useful, and I thank you for the reminder of its quality.
remark: I have read chapter 1 of A-M and worked most of the exercises, but in general there are just too many exercises in there for me not to get bogged down. This book's text also goes too fast for me. The proofs come so fast and briefly I don't have time to understand their implications. So I would need to discipine myself to read this book very slowly, stopping to think about all the slick proofs.
By the way, I could be wrong, but it seems to me the second to last sentence on page 31 of A-M is incorrect. They say there the A - algebra structure on the ring D is by means of the map a --> f(a) tensor g(a), whereas it seemed to me last time I read it was that it should be via a -->f(a) tensor 1 = 1 tensor g(a). Yes in fact this is forced by the very next sentence, giving the commutative diagram for the various given ring maps. The map they give is obviously not even additive, since f(a+b) tens g(a+b) does not equal f(a) tens g(a) + f(b) tens g(b). In my opinion that is the sort of thing that can happen when you go too fast and don't pause to explore the consequences of your statements, although these authors are so smart and knowledgeable, there seem to be remarkably few such errors.
I also have as introductory algebraic geometry books, Mumford's two books, yellow and "red", Hartshorne, Vakil, Miles Reid, James Milne, Mike Artin, Fulton, Walker, Shafarevich, Griffiths, Miranda, Griffiths and Harris, ACGH, Hassett, Bertram, Harris, Cox-Little-O'Shea, Semple and Roth, Fischer, Brieskorn and Knorrer, ... well I have a lot.
As a tip for reading Hartshorne's book, he himself wrote it after teaching several courses on the subject, some of which I sat in on. The first course was the basis for his chapter 4 on curves, and the second course was on surfaces, his chapter 5. Hence I recommend reading them in that order, i.e. start with chapters 4 and 5 and only then go back to 2 and 3 for background you may want to see developed in detail. Chapter 1 is independent of the others, a separate course on varieties and examples. In fact Hartshorne himself suggests starting in chapter 4, for "pedagogical" reasons, but only says so in the first paragraph of that chapter, which the reader may not have noticed until plunging haplessly into chapters 2 and 3.
I also celebrate the great effort Hartshorne has made to provide us a clear account of so many things, but his choice of just citing commutative algebra results without proof, does not work well for me. I prefer Shafarevich's model of actually proving the needed results as they are encountered, as he does especially in the first one - volume edition of his book, which I recommend highly. Mumford also tends to lose me in his red book on those occasions where he sends me to Zariski - Samuel for extensive background on fields, rather than just telling me the argument he needs. Zariski-Samuel is excellent, but the excursion means a big time sink for me.
Mumford is so knowledgeable and so succinct in his explanations that it is a great service for me when he just summarizes the proof of something, which he usually does. His redbook is the only place I know where one is told what is the relation between varieties over arbitrary fields, and the associated ones over their algebraic closure. After reading this, I was able to easily give a complete answer to a student question on stackexchange about what are the maximal ideals of R[X,Y], where R is the real numbers, whose full explanation had not been provided for some time (although correct answers to the more limited question actually asked had been given, and those people probably knew this as well).
Here is a tiny example of something I absorbed from Reid that I had not realized from any other source, although maybe I would have, had I read Bourbaki more fully. Namely, the primary decomposition theorem for noetherian rings says that every ideal in a noetherian ring, ( a ring in which every ideal has a finite number of ideal generators), can be written as an intersection of "primary" ideals. An ideal is prime, as you know, if when you mod out by it, you get a domain, i.e. a ring in which there are no zero divisors except zero. An ideal is primary if when you mod out by it, the only zero divisors are nilpotent. OK, the surprizing result is that even irredundant primary decompositions are not unique! I.e. the primary ideals involved are not always unique. BUT! those primary ideals that are minimal, ARE unique. Moreover, the prime radicals of both minimal and non minimal primary ideals are unique.
The geometric version of this says that every algebraic scheme in affine space, can be written as a union of irreducible algebraic schemes, and the maximal set theoretic components have a unique scheme structure, but those components that are contained in other larger components have a non unique structure. Nonetheless, the underlying sets of these component scheme are all unique, i.e. the radicals of the primary ideals are all unique. Now it had never dawned on me that the non uniquemess means that those primary ideals are not important. Namely it is the unique objects, namely the prime ideals occurring as their radicals that are important. Reid makes this clear by taking the Bourbaki approach to decomposition, showing that it is the "associated primes" of an ideal that should be focused on. I.e. one defines the associated primes, shows their uniqueness, and then proves that they are the same as the radicals of the primary ideals in an irredundant primary decomposition. Just a remark.
Summary: the primary ideals of embedded components are not important since not unique, rather the support of embedded components matter more. I never realized this before reading Reid. I could still be wrong of course, but I feel I have learned something.
