Explicit construction and computations on higher genus Riemann surfaces

Question

I'm learning about the higher genus ($g>1$) Riemann surfaces and I find it hard due to the lack of explicit examples. Specifically I'm interested in the basis of holomorphic forms, Abel map, prime forms and meromorphic functions etc. For a sphere and a torus everything can be made explicit, but at the same time also sort of trivial. For a sphere there is no Jacobian, while for a torus the Jacobian is isomorphic to the torus itself. I would like to be able to have some representation of, say, genus $2$ surface, so that I can write the holomorphic forms, compute the period matrix etc. Where can I find explicit examples?