Shooting a scene when there is fog or haze can create a very mysterious image but it can be really tricky to capture. Taking pictures on a foggy day or night is unlike taking pictures in clear weather because there is very little contrast and color saturation. And, while these conditions add to the moodiness of the scene they can be frustrating to photograph because sometimes a photo can look washed out and flat if you don't carefully select your settings. I'm no landscape photography expert but there are definitely some things that I've learned along the way that have been helpful.
1. Exposure compensation
Fog, which is essentially water vapor suspended in the air, makes the air more reflective to light so your camera will think (yes, your camera thinks) that it needs to decrease the exposure. In other words, your camera's light meter can't read the light conditions accurately. Since there is very little light in a fog scene and low light situations usually require longer exposure times, it might be helpful to have a tripod. However, if you don't have one with you, then your best bet is to dial in some positive exposure compensation (+1 or +2 stops usually works). Definitely take a few practice shots until you get the proper value.
Heavy fog lowers contrast and can make a scene look washed out in photos because it acts like softbox light does (it broadens and softens the light). You can add contrast by including an object or people or some other subject, in the image. Including a subject in the foreground will also add depth to the picture.
3. Shutter speed
Don't use your flash. You'll be tempted because it's dark and your camera will be slow but it will only wash out your image even more. Longer exposures (using a tripod or the image will be blurry) will give you a smooth look. Fast shutter speeds will give you that lovely clear fog scene and it will obviate the need for a tripod or some other steady surface. You can also try shooting at higher ISO but watch out for using ISOs so high that you get too much noise in your photos.
Compare the following three images all taken within 30 minutes of each other
A little washed out but the clear waves in the foreground make it stand out a bit.
The reeds in the front bring out the fog in the back.