For my own tastes I actually prefer the look of overcast days when using color film. Colors seem richer and not as garish as when photographed under bright sun. To each his own.
Shadows can be less harsh as well. You might have to concentrate more on close ups than on landscape scenes. These will probably look better in soft lighting than they would in full sunlight. One danger is that flat lighting might lead to flat-looking scenes. If it is only lightly overcast then there can be at least a little shadow for modeling. Take care to try an see how the subtle lighting affects the subject modeling.
Try to separate your subject from the background. I can't say this is always harder or easier under overcast conditions than under sunlight conditions, but I think it tends to be a little harder.
Soft lighting of an overcast day will be better for people pictures than hard sunlight, unless it's raining. However, even if it is raining it might lead to unique ways of looking at your subjects.
Obviously, exposure is going to be an issue, but the lower light levels will lead you to open up the aperture, which can be better at throwing the background out of focus, so choose subjects where this is a desired effect.
Others have mentioned filters.
You probably want to avoid including the sky in most of your photos, but in some cases you might want to include some sky for mood setting or to set the scene of the travel photo so you remember some aspects of the travel (for memory-type photos at least). Also, in some cases the sky itself may hold interest, especially on those occasions when the weather is breaking.
For film choice you might favor a higher contrast film and/or one with greater color saturation, unless you intentionally want to go for more subtle colors.
Most of my comments are a obvious, a bit naive, and amateurish, but might serve as useful reminders.
