Fog can add a beautiful element in your photography and the results can be quite unique and etherial if it's handled well.
Now, some will tell you that photographing in the fog is difficult. I've even read somewhere that it can be "one of the most difficult things to photograph." But that's simply not true. You just need to understand a little bit about how fog effects your photography.
Probably the biggest challenge facing a photographer in the fog is determining the correct exposure. In any automatic mode, your camera will tend to underexpose the scene. The reason is that your light meter is wanting to expose for an "average" scene but fog is not average. In fact, fog is generally brighter that middle tone.
As a general rule, fog is about one stop brighter than middle tone. So, if you're using an automatic exposure mode, you need to dial in a little bit of exposure compensation. The exact amount depends upon how much of your scene includes the scene but it's safe to say it would be somewhere around +1/2 to +1 stop.
I like to shoot in manual mode and use a spot meter. I just point the spot meter at the fog and set my exposure so that the meter reads +1. The results are perfect.
Your camera needs to see some contrast in order to set focus. The problem is that fog reduces contrast. To make matters worse, foggy conditions can often mean darker scenes which just exacerbates the problem.
The solution is to just switch off the autofocus and focus manually. You can also close down the aperture to give yourself a little more depth of field. Just know that you may end up with fairly slow shutter speeds so a solid tripod is a real benefit.