Concerning Geometric measure theory and BV functions, I would recommend Functions of Bounded Variation and Free Discontinuity Problems (by Luigi Ambrosio, Nicola Fusco, and Diego Pallara), Oxford Mathematical Monographs. The first two-three chapters may be of interest to you and they contain (non-trivial) exercises.
Concerning Sobolev spaces, I would also recommend the well known book by Brezis, Functional Analysis, Springer.