thick fog can emphasise depth, simplicity, cold, loneliness and be surreal, mysterious and moody as well as making light rays more evident
look for a dominant foreground subject and carefully compose your scene
you will often need to increase exposure by around +1EV to avoid a dull grey fog - unless that is what you want - try experimenting with manual exposure and trial and error
AF on background subjects can be very difficult due to the lack in contrast - you may need to manually focus
experiment with colour temperature (but lave a little blue if you want more realism) or convert to B&W - hence it is best to shoot in RAW file
don't bother with polariser filter as it will add little and just give more camera shake by forcing longer exposures
keep the front on the lens dry and clean from condensation
colourful subjects in the foreground can create a nice colour contrast to an otherwise low contrast scene
fog also allows opportunity to create beautiful, simple silhouettes, especially at sunrise or shooting into artificial lights to provide some backlighting
try working with the fog for the beauty of simplicity that thick fog provides
shooting in fog with sunrise coming through can create a wonderful effect
keep shutter speed to less than 1 sec if you wish to retain texture in the fog
remember fog density can change quickly but our eyes don't often detect this change well - but it can have a dramatic change to the photograph by allowing more of less of the background to be visible
in post-processing, consider increasing contrast to give more drama and consider using gradient filters to darken the fog and highlight your subject (if it is not a silhouette)