In Voevodsky (and Deligne): Lectures on cross functors, there is a discussion of the formalism of the four Grothendieck operations. This is maybe superseded by Ayoub’s work. However, the lectures have some nice brief background info on the formalism (for etale sheaves), including PD and the four kind of (co)homology, indicating where there are difficulties. There is not really any material on motivic homotopy theory, it just says that such applications will be given “later”.
Voevodsky’s Nordfjordeid lecture - supernice (Voevodsky folder). Working over a field unfortunately. Basic constructions, of SH etc. Brief discussion of Thom spaces and homotopy purity. Cohomology theories: the motivic EM spectrum, KGL, MGL, claim that the notions of orientation and FGLs have direct analogs for P1-spectra. The slice filtration (great intro), update on Open problems paper. The zero-th slice of the unit spectrum is HZ, this is known for fields of char zero. Brief discussion of AHSS. Appendix on the Nisnevich topology, Nisnevich descent, and model structures.
nLab page on Stable motivic homotopy category