I'm interested where in the study of algebraic geometry, one really needs the full theory of schemes for the first time? Are there any results about varieties where using Hartshorne Ch1 type of machinery gets very tedious, but where schemes make the argument natural? I'm interested in results that are easy to express in the Hartshorne Ch1 language, but where the proofs are not (without major difficulties) and why schemes are the natural setting where to prove this.
I've been reading in parallel Ch1&2 of Hartshorne and it seems very difficult to gain any appreciation of the theory of schemes.