Regular road shoes are not appropriate. Try to choose trails with no stream crossings. Even if the streams don't flood, they may prove difficult to cross. Drink water or sports drinks as you run, to avoid becoming dehydrated. Take care when running up or down steep hills.
Rain can make an otherwise easy ascent or descent slippery and dangerous.
Slow down on muddy or slippery sections. Beware of slick rocks, roots and grassy sections of the trail. Stay on the trail, even if it's muddy. Straying from the trail leads to unnecessary erosion and injures vegetation. Dry off immediately after your run.