Here are the things that you can do with the modular form $f$ corresponding to an elliptic curve $E$:
(a) Determine the number of points on $E$ mod $p$ by computing $a_p(f)$ (easy
for smallish primes via modular symbols computations).
(b) Compute (perhaps with some effort) a modular parameterization of $E$,
and then, by evaluating this at Heegner points, find a point of infinite order
on a twist $E_D$ of $E$, in the cases when this twist has rank one.
(c) Compute whether or not $L(E_D,1) = 0$ for every twist $E_D$ of $E$,
via modular symbols. If you grant BSD, this tells whether or not the
twist $E_D$ has infinitely many points.
I'm not sure what other facts about $E$ you are expecting to get. What is it
you would like to know about an elliptic curve in any case? For most people,
the rank (and especially whether or not it is positive) is the main thing, and
conjecturally this is what you can get from the $L$-function of $E$, which is
essentially inaccessible without modular forms, but is highly computable once
you know $f$. (And not just for $E$, but for all its twists.)
Maybe the other thing you might like to know is Sha of $E$, but this is not
proven to be finite in general. Nevertheless, modular forms can sometimes
be used to witness non-trivial elements of Sha. (Read about the theory of
``the visible part of Sha'', by Cremona and Mazur.)
Schemes play an enormous role in all the modern theory of elliptic curves, and have done so ever since Mazur and Tate proved their theorem that no elliptic curve over $\mathbb Q$ can have a 13-torsion point defined over $\mathbb Q$.
For some additional explanation, you could look at this answer. But bear in mind that theorems on the classification of torsion, while fantastic, are just a tiny part of the theory of elliptic curves, and a tiny part of how schemes are involved. One of the most important theorems about elliptic curves is the modularity theorem, proved by Wiles, Taylor, et. al. twenty or so years ago, which implies FLT. These arguments also depend heavily on modern algebraic geometry.
Also, the proof of the Sato--Tate conjecture.
Also, all current progress on the BSD conjecture.
The underlying point is the the theory of elliptic curves is one of the central topics in modern number theory, and the methods of scheme-theoretic alg. geom. are among the central tools of modern number theory. So certainly they are applied to the theory of elliptic curves
On the other hand, you won't find it so easy to synthesize your reading on the arithmetic of elliptic curves and your reading of scheme theory. For example, even Silverman's (first) book, which is quite a bit more advanced than Silverman--Tate, doesn't use schemes. Some of the arguments can be clarified by using schemes, but it takes a bit of sophistication to see how to do this, or even where such clarification is possible or useful.
Hartshorne has a discussion of elliptic curves in Ch. IV, but it doesn't touch on the number theoretic aspects of the theory; indeed, Hartshorne's book doesn't make it at all clear how scheme-theoretic techniques are to be applied in number theory.
With my own students, one exercise I give them to get them to see how make scheme-theoretic arguments and use them to study elliptic curves is the following:
Let $E$ be an elliptic curve over $\mathbb Q$ with good or multiplicative reduction at $p$; then prove that reduction mod $p$ map from the endomorphisms of $E$ over $\overline{\mathbb Q}$ to the endomorphisms of the reduction of $E$ mod $p$ over $\overline{\mathbb F}$ is injective.
The proof isn't that difficult, but requires some amount of sophistication to discover, if you haven't seen this sort of thing before.
Finally:
None of the results on torsion on elliptic curves over $\mathbb Q$, or modularity, or Sato--Tate, or BSD, will be accessible to you in the time-frame of your masters (I would guess); any one of them takes an enormous amount of time and effort to learn (a strong Ph.D. student working on elliptic curves might typically learn some aspects of one of them over the entirety of their time as a student). I don't mean to be discouraging --- I just want to say that it will take time, patience, and also a good advisor, if you want to learn how schemes are applied to the theory of elliptic curves, or any other part of modern number theory.
Best Answer
hunter's answer is a reasonable one, so I will just elaborate on some details. I'll name the books respectively Neukirch $1$, Neukirch $2$, Diamond Shurman, Silverman, and Advanced Silverman.
First, I am assuming you mean Neukirch's "Algebraic Number Theory"? I don't think he has written a book with the title "Algebraic Geometry".
Now, there are two clear sequences in that list of books, namely $$\text{Neukirch 1} \longrightarrow \text{Neukirch 2}$$ $$\text{Silverman} \longrightarrow \text{Advanced Silverman}$$ By this I mean that the book on the right presumes knowledge from, and is generally more advanced than, the book on the left. This doesn't mean you have to finish the book on the left from cover to cover to begin the one on the right, but I would advise getting through some introductory chapters of the left before opening the one on the right.
Diamond Shurman is a great book, and gives an introduction to most of the advanced concepts they use. The story they tell is, in very coarse terms, how rational elliptic curves correspond to modular forms. One of the very important ingredients of this is $L$-functions.
Note that there are also intersections among the books. to mention a few; modular forms and functions enter both Silverman and Advanced Silverman, while having a major role in Diamond Shurman. Varieties, including elliptic curves, are also part of Diamond Shurman while being the central topic of both Silvermans. Neukirch 1 devotes much of the latter part on class field theory, which is involved in (at least) the chapter of complex multiplication of Advanced Silverman.
However, in my experience none of these intersections are too serious - whenever you will encounter a topic for the second time, it would be with a different angle, and you will be able to skip past parts you know.
Just reiterating what hunter said, in math it is often wise having an eye on where you are going and trying to figure out what you need, in contrast to just reading textbooks from cover to cover. Opening an advanced book also often gives perspective and direction when studying the more basic material. That being said, here is my reading guide:
From there on you will have a solid grasp on the basic theory, with views into the advanced theory. You should by then also have looked at where you want to go, and be able to take informed decisions as to what to read next from then on.