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The question reminds me of the time when I was studying mathematics. I had attended a course on algebra (some basic theory of group, rings, categories, loads of Galois theory and some valuation theory, which was the main area of research of the lecturer). Commutative algebra was the first subject I studied on my own, because I wanted to attend a course on algebraic curves, and because I needed some commutative algebra for my diploma thesis. This is not a direct answer to your question, but maybe some thoughts from that time are of use...

Literature: As a student I found Bourbaki, Nagata (local rings) and Matsumura (Comm. rings) too difficult as a starting point. I liked the book of Atiyah-MacDonald and I loved the book "Kommutative Algebra" of Brüske, Ischebeck, Vogel. The Brüske-Ischebeck-Vogel book is out of print and available online http://wwwmath.uni-muenster.de/u/ischebeck/SkriptBrskeIschebeckVogel.pdf. If I had to teach a course on commutative algebra, then I would surely have this book on my desk again. Unfortunately it is written in German, but nevertheless it might be worth to have a look...

Free, projective and flat modules: I remember that I needed them for my thesis project and my thesis contained a section summarizing these things. I think at that time I was fine with the account in the BIV book. They define M projective iff $Hom(M, -)$ is exact, flat iff $M\otimes -$ is exact and injective if $Hom(-, M)$ is exact. I found this quite natural at that timesa first reading, and later on the step to $Ext$ and $Tor$ was quite natural as well. (Maybe I was influenced a bit by the fact that categories and functors were always in the air in Munich at that time and were mentioned in my algebra course.) Having these definitions at hand, one can directly go into the proof of theorems comparing these classes of modules...

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The question reminds me of the time when I was studying mathematics. I had attended a course on algebra (some basic theory of group, rings, categories, loads of Galois theory and some valuation theory, which was the main area of research of the lecturer). Commutative algebra was the first subject I studied on my own, because I wanted to attend a course on algebraic curves, and because I needed some commutative algebra for my diploma thesis. This is not a direct answer to your question, but maybe some thoughts from that time are of use...

Literature: As a student I found Bourbaki, Nagata (local rings) and Matsumura (Comm. rings) too difficult as a starting point. I liked the book of Atiyah-MacDonald and I loved the book "Kommutative Algebra" of Brüske, Ischebeck, Vogel. The Brüske-Ischebeck-Vogel book is out of print and available online http://wwwmath.uni-muenster.de/u/ischebeck/SkriptBrskeIschebeckVogel.pdf. If I had to teach a course on commutative algebra, then I would surely have this book on my desk again. Unfortunately it is written in German, but nevertheless it might be worth to have a look...

Free, projective and flat modules: I remember that I needed them for my thesis project and my thesis contained a section summarizing these things. I think at that time I was fine with the account in the BIV book. They define M projective iff $Hom(M, -)$ is exact, flat iff $M\otimes -$ is exact and injective if $Hom(-, M)$ is exact. I found this quite natural at that times. (Maybe I was influenced a bit by the fact that categories and functors were always in the air in Munich at that time and were mentioned in my algebra course.) Having these definitions at hand, one can go into the proof of theorems comparing these classes of modules...