Reference request: sheaf cohomology after Bott & Tu

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Undergrad here. I want to understand sheaf cohomology. (Not sure if this is relevant, but I also want to understand Étale cohomology.)

To do this, I've been studying de Rham cohomology from Bott and Tu, to lead into Cech cohomology, which I think is one potential route to general sheaf cohomology(?) It looks like B&T covers this briefly. I'll be trying to hit the following sections:

  • Section 10, presheaves and Cech cohomology
  • Section 13, monodromy

But past that, it looks like I'll need to find other sources for studying sheaf cohomology. So, my questions:

  1. For my purposes, are there other sections (past section 6) of B&T I ought to read?
  2. What is a good 'next read' for understanding sheaf cohomology?
  3. Is it advisable to study sheaf cohomology before etale, or are the two more or less independent?