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Perhaps the main thing beginners should learn about schemes is that they are needed. I.e. schemes should be motivated. In books which try to restrict to varieties such as Shafarevich's BAG, schemes still raise their heads sometimes unnoticed. E.g. Shafarevich states in chapter I sections 4.4 and 6.4 that the set of hypersurfaces of given degree in a given projective space are parametrized by a projective space, which is not true unless one considers more than the variety defined by a polynomial.

If one is guided on what to include by the section headings of chapter 2 of Mumford's red book, in addition to fields of definition and the functor of points, one finds there a section called specializations, which also contains one of his exotic illustrations.

Even in a classical book like Walker's algebraic curves, schemes arise when studying singularities. The tangent cone to a cuspidal plane curve requires more structure than a variety. Even the fundamental theorem of algebra does not count the roots of a polynomial correctly unless multiplicities are considered.

Some of these examples require only cycles or divisors rather than schemes, but more general tangent cones should provide more general schemes. One can also consider the problem of varieties varying in families and try to fill in something over the limit point of the parameter space. Sometimes non reduced objects will force themselves on us.

The best motivation for differentials may be learning the classical Riemann Roch theorem for curves.

Of course this is probably obvious and taken for granted by most people, but it seemed worth mentioning as a guide to choosing first examples of schemes. I.e. we should not take schemes for granted and choose what to teach based solely on the needs of experts, but we should assume that schemes may be quite strange to beginners and spend some effort showing that they are natural.

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In reference to why the spec functor is a natural thing, (low tech answer): isn't this essentially what the nullstellensatz says? Or rather it generalizes the nullstellensatz. I.e. spec is a good thing because it lets you make a construction that gives you some "geometry" associated to a given ring.