Plough or Direct Reining
The most basic rein aids help you tell your horse the direction to turn, when to stop or back up. As you advance in your riding skills, you'll learn to use rein aids in more subtle ways to cue your horse. But for now, you need to learn to hold the reins correctly so that you can convey your signals effectively and you don't create any bad habits that might confuse your horse or make it uncomfortable.
For plough or direct reining, most often seen in English style riding, you will be holding one rein in each hand. Pick up the reins so that each rein sits between your little and ring finger. The rein will lay across your palm and come out of your hand over your index finger. Your thumb should be upwards, so the buckle end of the rein comes out the top of your loosely held fist. If you were to draw a line across your knuckles the line would be about 30 degrees above horizontal. You don't want your thumbs to point straight up, and you don't want your hand to be flat across. Don't grasp too tightly. This will tire your hands and make your rein aids stiff and heavy for your horse. If you grasp the reins too loosely, the reins will slide through your fingers and your aids will be ineffective.
Your elbows, forearms, wrists, and hands should be in a straight line so there is a direct line from your elbow, down the reins to the horse's mouth. As the horse moves its head and neck as it travels, you hands should follow the movement. Your body should not rock back and forth, but the movement should come from your arms and shoulders, rather like reaching forward, without bending your body forward. Your wrists should not rock, but maintain the straight elbow to bit line.
As you ride you should feel light, consistent contact with your horse's mouth. You don't want to be too heavy handed, or as I mentioned, hold the reins to loosely. If you constantly pull on your horse's mouth, it may learn to ignore your cues or react by tossing its head or rooting to relieve the pressure. When you hold the reins too loosely, it's easy for your horse to pull the reins out of your hands and your rein aids will be ineffectual.
Western Neck Reining
It's very handy, whether you ride English or western, to have a horse that neck reins. When you neck rein, you'll hold both reins in one hand. Traditionally, the reins will be held in your left hand. This is traditional because it left a cowboy's dominant hand free (usually the right one) to rope cattle, manipulate gates and do other work. If you choose to hold the reins in your right hand or left won't matter if you're just going to be riding and not actually working cattle.
Rommel reins (reins that are attached at the ends) are held in a vertical fist, with the reins coming up through the bottom of the fist, through your hand against the palm and the rein end coming out of the top of the fist and back over the thumb. Just as with direct reining, you want your thumb pointing up, but at 45 degrees from horizontal rather then 30 degrees. Your free hand may carry the end of rommel reins, leaning against your leg, in a way that keeps you sitting squarely in the saddle
Another method with split reins is to hold the reins with your hand in the same position, but with the free end of the reins coming out the bottom of your fist, past your little finger. You may split the reins, holding one between your index and second finger. The ends of the reins dangle down the horse's shoulder. This is the method I learned long ago.
Western riders usually ride with less contact than English riders, so you should just feel the weight of the reins, rather than contact with the horse's mouth. Pulling too hard will cause the same problems as heavy handedness with direct reining. In fact, because horses ridden western are often wearing a curb bit, more severe problems like rearing can occur.
Gaming or roping reins are shorter than regular rommel or split reins and are one piece end to end. These are carried similar to split reins, with all the fingers closed around the reins. Gaming reins are shorter so that there is less chance of hands or props getting tangled in the reins.
Different disciplines have specific rules, so if you're going beyond pleasure riding and becoming involved in competition, it's best to find out exactly what the expectations are. For example, in some types of western classes, splitting the reins between your index and second finger is considered incorrect. Read the rule book and have your coach catch any habits that may disqualify you.