For Your Feet
A good pair of running shoes is essential. If you are planning on doing some serious trail running, investing in trail specific running shoes is well worth your while. Unlike road running shoes, trail shoes are made to weather the harsh terrain that you will find on the trails. The out soles on trail runners are made of a more durable rubber that is usually a bit tackier than road running shoes. This will grip rock and other slippery terrain that you will encounter on the trail. When running on the trail, you don't need as much cushioning as on the roads, the dirt will do that for you. So trail runners can be built with a lower profile and a firmer mid sole. This will give you a better feel for the trail, allowing you to be more nimble and quick. Also, trail shoes have more durable uppers that are made to withstand the rigors of the great outdoors.
Running Wear
Gone are the days when you would throw on a cotton Tee-shirt and shorts and head out for a "jog." Technical running wear is far superior to old school gym sweats. Lightweight and moisture wicking, these fabrics nearly magically keep you cool in the heat and warm in the cold. And very importantly, the moisture wicking properties of technical running wear help prevent chaffing on long runs, a most uncomfortable experience. You should also add a waterproof running jacket to your wardrobe, preferably one that packs small and can be tucked into a running pack. It is very important that you invest in some moisture wicking running socks. There are even some made especially for trail running that are extra durable and wicking. Cotton gym socks will only hold in moisture and cause blisters.
Accessories
Trail running is an adventure, and for every adventure you need your gear. You need to make sure that you are hydrated so packing your liquids is essential. You can choose from hip packs with water bottles or back packs with bladders for your liquids. Backpacks are better for longer runs of more than an hour. These packs are often like mini backpacks and have many zippered compartments for energy gels, maps, medications and more. If you are going out for a shorter jaunt you can just strap on a hip pack with a water bottle or use a hand-held bottle with a strap.
Since you'll be out in the elements for some time on trail runs, you'll probably find that wearing a hat and sunglasses is more comfortable. Hats with brims protect your head and face from the glaring sun and sunglasses protect your eyes from the sun's damaging rays.