I am interested in a description of orbits of the natural $GL(V)$ action on $V^{\otimes d}$. I know this is a classical problem but I tried to find some "good" reference and I couldn't. I'm also interested in orbits of the natural $S_d\times GL(V)$ action on $V^{\otimes d}$. I'm assuming characteristic $0$ but references in finite characteristic are also welcomed.

See, for instance, Chapter 5 of Goodman and Wallach's book "Symmetry, Representations and Invariants". Another good reference is Procesi's "Lie Groups: An approach through Invariants and Representations". 

