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Quick-ish Introduction to Schemes in AG

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*The Geometry of Schemes* by Eisenbud and Harris may be what you're looking for. Don't think they have a lot of material on cohomology, though.
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First 5 chapters of Qing Liu (around 200 pages I guess), you can skip some sections and propositions. It somehow relies on exercises but there’s little difficulty even if you don’t work through them.
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Gathmanns Algebraic Geometry lecture notes. They start from the usual definition of varieities over fields and then later on introduces schemes to rework the concept. Additionally sheaves and sheaf cohomology are covered
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Peter Scholze wrote up a rather concise and modern set of notes on schemes, which he built from the ground up in that course.

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