First please don't be bluffed by those fancy terms coined by computational scientists, and don't worry about preconditioning or conjugate gradient. The multigrid method for numerical PDE can be viewed as a standalone subject, basically what it does is: make use of the "information" on both finer and coarser mesh, in order to solve a linear equation system(obtained from the discretization of the PDE on these meshes), and it does this in an iterative fashion.
IMHO Vassilevski from Lawrence Livermore national laboratory puts up a series of very beginner-oriented lecture notes, where he introduced the motivation and preliminary first, how to get the $Ax = b$ type linear equation system from a boundary value problem of $-\Delta u = f$ with $u = g$ on $\partial \Omega$, what is condition number and how does it affect our iterative solvers.
Then he introduced all the well-established aspects of multigrid: what is the basic idea in two-grid, how do we do smoothing on the finer mesh, and error correction on the coarser mesh, V-cycle, W-cycle, etc. Algebraic multigrid(the multigrid that uses information from mesh is often called geometric method), also the adaptive methods are covered too.
Some example codes for Poisson equation can be easy google'd. If you got more time, this book has a user-friendly and comprehensive introduction on this topic together with some recent advancements.