Really all you need is the appropriate clothing for the conditions you will be running in. That being said, I have seen the full spectrum of clothing on people who trail run, some guys can wear short running shorts and a baselayer T in temps down to the 40's, it really all depends on the person. Once you start running your body temperature will rise quickly.
1) 16-24oz. is a typical amount of water to carry for a shorter run lasting 1-2hrs. 20oz. every hour is a good place to start, but I rarely need that much on a run lasting less than 2 hours. Of course if it is blazing hot then I may bring 48oz. for a run lasting 2 hours. (I live in NM)
* Like the above poster, how you carry is up to you, waistpack, handheld, hydration pack.
2) Appropriate shoes for the terrain you are running and the weather conditions. If you run in dry or warm regions opt for something breathable and quick drying to aid in blister prevention and to prevent your feet from getting hot.
3) Breathability is key with trailrunning, finding clothing that transfers moisture away from your skin and aids in evaporation is important. There are several companies that make clothing for high exertion activities like trail running that provide a range of light/heavy insulation and appropriate temperature regulation. The Patagonia capilene line of clothing is an excellent start for finding a good baselayer for warm to cold conditions.
A breathable windshell is a nice feature on chilly runs, they often don't weigh more than 6 oz. and can be easily balled up and stored in a waistpack or carried in hand. I bring a windshell on chilly morning runs before the sun comes up, it helps prevent the wind from robbing warmth.
Headwear, consider the conditions, make it breathable.
Legwear, it is up to you, but the options are endless, loose tights, compression tights, 3/4 tights, pants, shorts.
3) What personal items you carry is up to you and your required level of safety. I rarely carry anything besides my car key, a gel or 2, and perhaps my drivers license if I'm going on a run lasting more than 2-3 hours out on the trail. Some people run with cell phones, it is up to you and how important getting in contact with or being contacted by the outside is to you. Letting someone know what trail you are running on, when you left, and when you expect to return is good practice and only takes a second and could save your life.