There are some old tutorials that show how to do concrete calculations with these objects with Macaulay. They were an immense help to me when learning algebraic geometry, because they showed me how all this sheaf blah blah blah connected to actual concrete polynomials.

Try these two: https://faculty.math.illinois.edu/Macaulay2/Book/Computation... https://faculty.math.illinois.edu/Macaulay2/Book/Computation...

There's more in the book "Computations in algebraic geometry with Macaulay 2," which is free online. See also Schenck's "Computational Algebraic Geometry."

edit: Also, topos theory is definitely not the way to go here. Getting your hands dirty with algebraic curves is. See e.g. Griffiths' Introduction to Algebraic Curves.