I have found Kenji Ueno's book Algebraic Geometry 1: From Algebraic Varieties to Schemes to be quite satisfying in introducing the basic theory of schemes. Well, to be fair, this is only the first in a series of three books on the subject by the same author. So this first volume basically just develops the definitions of an affine scheme first and then of a scheme in general by "pasting" together affine schemes. It does not go into cohomology and more advanced stuff, which is the subject of the other two books.
However, what I really like is that he motivates very carefully the passage from the definition of an affine algebraic variety as an irreducible algebraic set in an affine space $\mathbb{A}_{k}^{n}$ to the definition of an affine variety using schemes, which is where you are having some trouble. What he does is that he starts by doing some algebraic geometry in the classic sense, that is, over an algebraically closed field $k$, in the first chapter of the book.
Then the author proves that there is a correspondence between the points in an algebraic set $V$ and the maximal ideals of its associated coordinate ring $k[V]$, where a point $(a_1, \dots , a_n) \in V$ corresponds to the maximal ideal of $k[V]$ determined by the ideal $(x_1 - a_1 , \dots , x_n - a_n) \in k[x_1, \dots, x_n]$, that is, a correspondence between the points in $V$ and the "points" in the maximal spectrum $\text{max-Spec}(k[V])$ of the coordinate ring $k[V \, ]$.
Then Ueno goes on to define an affine algebraic variety as a pair $(V, k[V \, ] )$ where $V$ is an an algebraic set. But he then makes the argument that one can go a little bit further and consider the pair $( \text{max-Spec}(R), R )$ where $R$ is a $k$ algebra. But here Ueno arguments that if the original intention was to study the sets of solutions of polynomial equations, then where is the geometry and where are the equations hidden if an algebraic variety is defined as this pair $( \text{max-Spec}(R), R )$?
The interesting thing is that if the $k$ algebra $R$ is finitely generated over $k$ then
$$ R \simeq \frac{ k[x_1 , \dots , x_n] }{I}$$
so that as a consequence
$$ \text{max-Spec}(R) = V(I)$$
so that again you'll have some equations (this is all done and explained in detail in the book). So then the author (re)defines an algebraic variety over an algebraically closed field $k$ (remember that he is doing everything in the classic sense) as a pair $( \text{max-Spec}(R), R )$, where $R$ is a finitely generated $k$ algebra.
And then at the end of the first chapter the author motivates the need for a more general theory, for example having in mind the needs of number theory, because since everything was done in the context of an algebraically closed field, then the arguments don't work for the fields (and rings) of interest in number theory. In particular, it is noted how an extension of the definitions to include these cases would need to take into account not only the set of maximal ideals, but the set of all prime ideals.
Then chapter two develops first some properties of this set of prime ideals, or prime spectrum of a ring, making it into a topological space with the Zariski topology... and then defines the necessary things in order to be able to define an affine scheme and a scheme (I mean, the concepts of a sheaf of rings, a ringed space, etc).
It is not a short story of course, but again I prefer this type of approach at first, than having to deal with an unmotivated (and difficult) definition that strives for great generality but I have no idea of where it comes from and what is its purpose.
Note that the book that Arturo recommended is great also but it assumes you already know some algebraic geometry and its level is higher than Ueno's book.
You should take a look at it and see if you like it, the book has a fair amount of examples and some exercises interspersed within the text also. You'll have to study from other sources as well but I believe that this book does a pretty good job at motivating the abstract definitions.
I hope this helps at least a little.
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.
Best Answer
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.