I am starting to learn about sheaf cohomology but, since I am interested not only in algebraic geometry but also in complex geometry, differential geometry and analytic geometry, I wonder if sheaf cohomology on schemes, manifolds and analytic spaces are indeed different or it I can just study it in (locally) ringed spaces and then have most results valid in whatever context I wish.
If the "natural" environment of sheaf cohomology really is in ringed spaces, what would a good reference for it?
On
For a relatively general algebraic approach, I'd say
Try to understand Chapters 1.1-1.3, 1.6 and Chapter 2 of Weibel's Introduction to Homological Algebra first. Most of that is so general, you can apply it in many different contexts.
Get started on sheaf cohomology from whatever source you like. For me, this was struggling through Hartshorne, mostly because my supervisor knows the book by heart and is best able to help me when I have questions from there.
Do boatloads of exercises rigorously.
Warning: Be ready to fill in lots of missing details if you want to believe the proofs in Hartshorne.
My impression is that sheaves aren't really so important in geometry outside of algebraic or complex geometry - folks often prefer bundles and/or other constructions, or the sorts of problems sheaves are good for solving aren't really aligned with the big problems one wants to solve in that field (cf this MO question for more explanation). Your guess that the natural environment for sheaf theory is locally ringed spaces is correct, and everything you mention can be treated as a locally ringed space when equipped with the appropriate sheaf of functions.
As for references which treat sheaf cohomology on locally ringed spaces, some basic references would be Godement's Topologie algébrique et théorie des faisceaux and Iversen's Cohomology of sheaves. Stacks Project also presents a good amount of material in this area too.