I created fog from 2 components. Horizontally aligned, big, slow particles with shader panning two noise textures at different speeds (something similar to this) + regular billboard, cloudy, alpha blended particles to add some volume to the fog (and receive some light from the lamps).
Rain has more elements to it, but I think most important are: rainsheets to reduce number of required particles, and a bunch of simple meshes with panning, transparent water textures to simulate water flowing down the objects (similar to panning fog shader, but with additional animated layer to create the edge where water is falling from the object surface).
There is also a bit of post processing (bloom, colour corection) on both cameras.