I want to learn about different cohomologies in algebraic geometry with a view towards understanding the idea of motives. I'm more interested in starting with a general overview or history. The main thing I am confused about is that there seems to be two very different meanings of the word "cohomology". In Hartshorne, we are introduced to what I will call "coherent cohomology" via the derived functor of the global sections functor. Then we find that this agrees with Cech cohomology in most useful cases.
But it seems like in modern algebraic geometry, the term "cohomology" refers to Weil cohomology theories. Can the derived functor cohomology be made into a Weil cohomology? Does every Weil cohomology arise from a derived functor? Is the derived functor cohomology of Hartshorne worth studying these days, or is most work done in terms of Weil cohomology theories, or are they largely unrelated? In general I am trying to understand how etale, or crystalline cohomology, for example, relate both mathematically and historically to what I have learned from Hartshorne.