I am far from an expert in algebraic geometry, but I think that there is something that really should be said in answer to this question.
Namely, there are at least two big reasons one might introduce a new theory--one is to answer old questions, and the other is to ask new questions (often, of course, motivated by the old classics). Of course, scheme theory was introduced to a large extent for the first reason. But I, at least, think the greatest virtue of the theory is in the second area.
Almost every question we could have asked about complex varieties, or at best about varieties over perfect fields (or maybe dvrs, in the lead-up to the development of schemes), can now be asked about schemes over arbitrary rings (or non-affine bases). We can see the geometry in diophantine equations; we can use cohomological tools to answer arithmetic questions. We can analyze deep algebraic structures of commutative rings in a geometric way. See e.g. Minhyong Kim's answer here.
I think this is what Kevin Buzzard's comment is getting at, in an extremely pithy way, in his comment about Fermat's last theorem; one might make a similar comment about, say, Falting's theorem. In order to even ask the right questions to approach these classics, we must have access to geometry over all sorts of supposedly pathological bases: imperfect fields, or even non-Noetherian rings. 100 years ago, who could have even imagined the statements of theorems about modular curves, for example, which seem totally natural today?
So back to your question: what elementary questions can be addressed using scheme theory? I guess I would say: any question about families, all of arithmetic geometry, any question about varieties over $\mathbb{C}$ you might be interested in over another base, any application of cohomological methods from the analytic theory (e.g. Riemann-Roch) you want to generalize, almost any problem where moduli spaces come up, etc.