My suggestion is you focus on special groups. For instance random walks on solvable Lie groups. The best reference is the doctoral thesis of Tianyi Zheng in Cornell University. You can find the pdf fine here. Also on lie groups you can find this nice paper here. About random walks on nilpotent lie groups you can find the best reference in a doctoral thesis here
About random walks on finite groups, the best can be the note of Laurent Saloff-Coste entitled Random Walks on Finite Groups. You can find the pdf file here