I'm trying to learn the basics of elliptic curves without having to learn schemes. However, the notion of a curve keeps throwing me. Perhaps I'm off the mark here, but it seems to be the case that algebraic geometers like to think of curves as being more than just their sets of points. In light of this, I was thinking that reasonable non-scheming definitions would be:
Idea. Let $k$ denote a field. Then an affine curve over $k$ is a non-zero prime ideal of $k[x,y].$ And a projective curve over $k$ is a non-zero homogeneous ideal of $k[x,y,z].$
Is this basically correct? If not, some guidance would be nice, as I'm pretty lost.
What you've defined are (irreducible) affine curves equipped with a choice of embedding into the affine plane $\mathbb{A}^2$, resp. (irreducible) projective curves equipped with a choice of embedding into the projective plane $\mathbb{P}^2$ (and your projective definition also needs a primality condition or else you allow ideals which aren't radical); (irreducible) plane curves for short.
(Although there's a subtlety in your definition of projective curve, which is that a morphism of projective curves is not the same thing as a graded morphism of homogeneous coordinate rings.)
Unfortunately, most curves are not plane curves: the genus-degree formula implies that an irreducible projective plane curve can only have genus a triangular number $0, 1, 3, 6, \dots$, so for example no curves of genus $2$ embed in the plane. It's a nontrivial exercise using the Riemann-Roch theorem to show that every elliptic curve is a plane curve.
Algebraic geometers do indeed like to think of curves as being more than their sets of points (if by "points" you mean $k$-points); this is absolutely crucial. For example, the affine curve $x^2 + y^2 = -1$ has no real points, but it has many complex points and hence should be distinguished from the empty variety over $\mathbb{R}$.
For many purposes you can get away with thinking of a variety over $k$ in terms of its $\overline{k}$-points together with the Galois action on them. However, based on your other questions it sounds like you want to understand kernels of isogenies between elliptic curves in characteristic $p$; these kernels can be non-reduced group schemes and so you really do need to learn at least a little bit of scheme theory to understand them. So far I personally have managed to get away with thinking only in terms of functors of points and never having to learn about locally ringed spaces.
A definition of affine curve along the lines of what you propose is that an (irreducible) affine curve is (the Spec of) a finitely generated integral domain over $k$ with Krull dimension $1$. You can give an analogous definition of projective curves although you will again run into trouble when trying to define morphisms.
Anyway, if you want more help it would be helpful to be more specific about what you mean by "the notion of a curve keeps throwing me."