First, helmets. If you are going for a mountain bike ride you should have a helmet, and it should be on your head. Taking your helmet off while climbing is just about as bad as doing the entire ride without it. I've gone down just as many times, if not more, while going up as I have while going down. Furthermore, it's not hard to figure out that one rider is almost always going up in a head on collision.
Second, a pump, and patch kit or a tube. You should always carry with you at least one way to fix a flat. Nobody likes the walk of shame back to the trailhead. I always carry a pump, a tube, and a tiny patch kit so I can usually fix any flat I have as well as help the poor fool who left theirs at home.
Third, tire pressure. One of the most common mistakes people who don't ride very often make, is heading out on the trail on tires that have lost most of their pressure. This is understandable, but it can very quickly lead to a pinch flat or a torn stem. Either way, riding off without checking your tire pressure can be a quick road to fixing a flat.