Starting on Monday I will be teaching a (first) graduate course on the arithmetic of elliptic curves. The two texts that I will be using are Silverman's Arithmetic of Elliptic Curves and Cassels's Lectures on Elliptic Curves.
The course does not have any algebraic geometry as a prerequisite. Some students have seen a little algebraic geometry or will be taking a first course in that subject concurrently; a few have seen a lot of algebraic geometry. But at least a few have never taken and will not concurrently be taking any algebraic geometry whatsoever. One of them asked me about this, and I confirmed that the course should still be appropriate for students like him.
If you want to learn about elliptic curves beyond the undergraduate level, you will need to start engaging with some rudiments of algebraic geometry: for instance, really understanding what is going on behind the group law on an elliptic curve requires (in my opinion, at least!) a discussion of the Riemann-Roch Theorem on an elliptic curve. However, elliptic curve theory is concrete enough and the algebraic geometric input is (at the beginning) limited enough so as to make it an excellent opportunity to learn some algebraic geometry from scratch. (I think you will get a taste of that subject faster by learning some elliptic curve theory than by learning commutative algebra, although of course the latter has an essential place in the long run.)
Further, Silverman's book is especially excellently written with respect to this issue: he puts all the algebraic geometry into the first two chapters. I would -- and will! -- recommend that you begin by reading through Chapter 1 on basic algebraic geometry: it is written with a very nice, light touch and mostly serves to introduce terminology and very basic objects. Then I would skip past Chapter 2 and come back to portions of it as needed in the rest of the text. For instance, if you've never seen differentials before, I wouldn't read about them in Chapter 2 until you get to the material on invariant differentials on elliptic curves in Chapter 3.
If it freaks you out to page past two chapters on algebraic geometry, than I would recommend starting with Cassels's text. He takes a more gradual, lowbrow approach to the geometric side, but he is just as much an arithmetic geometer as Silverman, so the approach he takes is quite compatible with a more explicitly geometric perspective which may come later.
I honestly think that these two texts are so excellent that you need look no farther. If it helps, many people around here can tell you that I am very fond of writing my own lecture notes for the graduate courses I teach. However, I wouldn't dream of doing so in this case: what Cassels and Silverman have already done is essentially optimal.
Take a look at Keith Conrad's expository articles here https://kconrad.math.uconn.edu/blurbs/ on Galois Theory and algebraic number theory. They are all wonderful. In particular, he shows you how to compute stuff using the tools you learn which is very essential in my opinion. Most of the topics on Galois Theory covered in his notes seem relevant to algebraic number theory, but I haven't read all of them in detail because I learned Galois theory in a college course. There is a fair bit of elementary algebraic number theory (at the level of Samuels's book mentioned by lhf above) which you will need to be fairly comfortable with before you move on to to class field theory. I think Keith's notes on algebraic number theory covers this elementary material well, with many examples of computations. Finally, I highly recommend his note "The History of Class Field Theory". I am certain you will be adequately prepared to read Lang if you work through some of Conrad's material.
For algebraic number theory, I also recommend Cassels-Fröhlich's "Algebraic Number Theory" and Cox's "Primes of the form $x^2 + ny^2$". James Milne's notes on algebraic number theory and class field theory, freely available on his website, are great. The other standard reference is Neukirch's "Algebraic Number Theory" which I personally really like. When you read about valuations, completions etc., I recommend the handouts from Pete L. Clark's course available here: http://alpha.math.uga.edu/~pete/MATH8410.html. Also, highly recommended are Serre's "Local Fields" and Iwasawa's "Local Class Field Theory" (the latter is harder to find, but I have a pdf copy which I am willing to share with you). You see from this that there are many good choices, and you will have to choose your own poison.
I should mention that there are different approaches to class field theory (discussed in the following link: https://mathoverflow.net/questions/6932/learning-class-field-theory-local-or-global-first), and I learned local class field theory through Lubin-Tate formal groups. Their short paper on this topic is a wonderful read. Luckily, Prof. Lubin frequents MSE so he may be able to recommend more sources.
I think that if you want geometric motivation for commutative algebra, you will need some knowledge of algebraic geometry (at least at the level of classical varieties) and DonAntonio's suggestions are great. However, you can learn the required commutative algebra as and when you need it. Certainly books like Cassels-Fröhlich prove most of the results of commutative algebra used in algebraic number theory along the way.
If you want to learn commutative algebra as a subject in its own right, take a look at Eisenbud's "Commutative Algebra with a View Toward Algebraic Geometry" which motivates the algebraic constructions using geometry. However, as mentioned in the previous paragraph, you may need some basic knowledge of classical varieties to understand the geometric motivations in this book. For this, I recommend Karen Smith's "An Invitation to Algebraic Geometry". Also, there is this new gem A term of Commutative Algebra by Altman and Kleiman, which is like an Atiyah-Macdonald 2.0. In particular, the authors mention that their aim is to improve some of the exposition in Atiyah-Macdonald using categorical language. The notes are really wonderful.
Despite these recommendations, I don't think you need so much preparation to read Lang. In particular, I came across the various texts mentioned in my answer as and when I needed to learn a particular topic.
Best Answer
Samuel's book is very usefull for the basics of algebraic number theory, i.e., Dedekind rings, splitting of primes, class group etc. with which you should surely be comfortable with at some point. Also because Elliptic curves gives you Dedekind rings as they coordinate rings. Furthermore, the book builds every result needed, giving you some quite useful example in commutative algebra and theorems stated in a narrower generality than in the AM. This may be helpfull for somebody who is self studying.
Then probably for everything else you need you can either pick Serre's Local fields or Neukirch. However if your purpose is only understanding the book by Silverman, then you can avoid the chapters on Class field theory and just read the first couple of both books, which cover the construction and classification of local fields. Class field theory is a beautiful subject, but is not needed and you may find yourself stuck in a rabbit hole.
Edit: I forgot to mention that another book which covers the same material as Samuel is Marcus's "Number Fields". It has the advantage of having plenty of exercises.