Shooting fog is somewhat tricky, so don't give up.
Not every fog is created equal.. You don't want it to be so dense that you can't make out a thing, and it should be thick enough to put a veil over more distant objects. The other thing you have to look for, is the light. When it's early in the morning the light has a warm colour that adds a delicate mood to the scene. Finally, for me to make a picture with fog work, I choose a scene with distinct fore- middle- and back grounds, and a not to wide focal length (e.g. 50mm)..
All in all, this means getting up before dawn, seeking out a location that has the appropriate subject material, and then hope for the proper weather conditions.
The image attached could have been taken with a $20 camera..