BACK-lighting, or very strong side-lighting, makes mist look its best. It helps if you can shoot it with sunlight streaming over the tops of a darker background, like deciduous trees. A good spot to shoot mist is at Horsetail Falls in the Columbia Gorge, where TONS of mist blows in the wind, from a waterfall and following the path of the creek to the northward direction, and in the afternoons in the springtime, sunlight streams over the tops of deciduous trees, or through the trees, which are dark, and the mist is brightly backlighted, and shows up well since the trees drop to basically almost-black.
Mist is basically colorless, so it shows up best against a strong colored background, as can be found near a lot of waterfalls, or ocean beach areas, where mist blows along with the wind currents. You do NOT want to under-expose mist or fog, or it looks bad.
FOG is different from mist, it really is. Fog almost always requires deliberate, substantial OVER-exposure from the light meter's suggested reading, and does not depend nearly so much for a darker background for it to show up well, because fog is so much bigger, and more all-enveloping a thing. In addition to over-exposing, it is often very helpful in fog to set the camera's tone curve to HARD, to make the degree of contrast very high.
In fog is a time when bracketed exposures, almost all "overs", can make the results look the right way more often than anything that is even remotely "under", exposure-wise. Fog is very sensitive as well to back-lighting or strong side-lighting conditions. When the sun is overhead, or high up, fog looks one way; when the sun i s low, and it is foggy, there can often be some interesting looks. As the fog LIFTS, the photo opportunities can just start appearing all over the place.