The walk is a Four-Beat gait, meaning that as the horse walks, four beats are heard before the sequence starts again. The sequence usually starts with the left hind leg. As this leg is about to land, the left foreleg moves. The same happens on the right side after the left has finished, which means we can count a definate even 1, 2, 3, 4 beat. If the beat is more 1, 2...3,4 then the horse is probably hurrying or being lazy.
Keep your weight in your seat bones, and ask the horse to walk forward by squeezing with your calves. As you feel the horse move forward, loosen off your reins slightly to allow the horse its head and the freedom to move. A tight rein would ask the horse to slow or stop. Remeber to keep your legs relaxed and long. Do not grip with the knees or thighs gently keep soft pressure with your calves as this will help you in stunt and other areas of riding (eg. Jumping and cantering), do not simply allow your legs to drop around the horse.
Remember to keep your heels down and toes in (toes in especially if hacking, don't want to get caught up on any trees) but this can be done by simply relaxing the legs and not restricting them from curving around the horses sides. As the horse walks, you will notice a rocking movement from left to right, as well as back and forwards. If you just relax your lower back you will feel it rocking too, your left hip rocks forward as the left hind leg moves and vice versa. As such you should tilt the appropriate side forward in time with the horse. Slowing your rocking restricts the horses movement, and most horses take that as an aid to slow down, and if you move faster the horse will speed up. Remeber to keep your shoulders level and back 'straight' (no leaning back or arching forwards) with your elbows and wrists in a straight line to the horses bit.
The trot is a two beat gait, meaning that only two beats are felt or heard before the sequence starts again. It is a diagonal gait, meaning that the diagonal legs move in pairs, for example the left hind and right forelegs will move at the same time. One diagonal will move, and dependant upon wether the horse is collecting, extending or working. The diagonal pair will move usually slightly before the first pair hits the ground, meaning there is a slight moment of suspension. In the posting trot you will have to rise on the right "diagonal" look at the outside leg every time it goes forward rise in your seat, it will help the horse keep balanced.
The canter is a 3 beat gait, however you can infact hear the fourth beat milaseconds after the third. It goes front left, front right, back right, back left. The front and back right often move in a lateral line. The rider should bounce out of the seat for 1 beat, and sit in the seat gently for another 1. Some horses have more of a "rocking horse" motion it means he/she will have a bigger range of movement and you will ride the same way.