I want to use the PETSc library to do some numerical work on finite element and parallel computing, but I wonder what I should know first to use these libraries. Could you give me some guiding information? Any answer of comment will be appreciated. Thank you.
PETSc is an outstanding linear algebra library. It also has modules for nonlinear solvers and, to some degree, for meshes and finite elements, but the latter aren't frequently used as there are much more comprehensive finite element packages out there. So, if you're interested in FEM calculations, my recommendation would be to go with one of the other packages (which oftentimes use PETSc internally to do their linear algebra computations).
If your goal is to do linear algebra, however, then PETSc is the right tool. As has been mentioned, the example programs help, but the manual is pretty decent to get an overview of what is available. You won't have to read the whole thing -- just stick with those parts you're particularly interested in.
Start with src/ksp/ksp/examples/tutorials/ex2.c or ex2f.F and work your way upwards.
PETSc is very easy to use. You just have to worry about local element level calculations (matrices, rhs etc.) and PETSc takes care of everything else (e.g. parallel assembly and solving).
Developing a parallel high performance FE code quickly is very easy with PETSc even if you don't have a strong background in scientific computing.
There is a slight learning curve and a PETSc way of doing things but you should be productive fairly quickly.