lesson of the leg
So you want him to walk faster? Will he walk faster on the way home to the cozy barn, his chow and his bud? First of all you have to determine his natural "walkout" speed. Some horses walk faster than others and it would't be fair to ask him to walk all the time at a pace that isn't his natural rythm.
However, if he is just being lazy and igoring your leg aid, then he needs the lesson of the leg. You have to reteach him what your leg aid means.
It means you have to put your leg on and MEAN IT! (it also means that you don't mindlessly bang away at his side every stride or you will dull him out over time). Your leg has to have an ON and an OFF and never the twain shall meet.
So, you put your leg on (I use one leg only , especially if the horse tends to balk back agains pressure.) Don't squeeze so much as "flutter" your ankle and calf against him. ONCE lightly, next Stronger and Last you take you whip and you bop, bop, bop in rapid succession right back where you put your leg on. Yes, it is sudden and shocking but it is meant to make an impression and wake him up! Guess what? He's gonna go forward like no tomorrow, and you'd better be ready to go with him. Do NOT pull him back down with the reins. You said FORWARD, so let him go forward at whatever gait he needs. Go with him for a bit and then ease him back down to a walk and act like nothing bad happened. Walk forward and put the leg on (lightly, harder and bop, bop with the whip!) If he repsonds at Light, or at hard, then don't use the whip. After one or two times, you just have to touch him to go. The secret is that you must go back to square one periodically, if he forgets the meaning of the leg on.
It is much better to be quick and firm than to be timid and nag. It is sometimes hard for us to do this as it causes our horses to react suddenly and we may be too scared to deal with that. If so, don't be too harsh with the whip, but at least wake him up by perhaps slapping your own thigh with a loud whack and most importantly, when you get serious and ask him to go forward, don't give up until you get a CHANGE in his walk, a real change, and then "coast" for a sec. and be sure that when he goes forward as per your insisitance that you dont' counter that with a "whoa" signal throug the reins.
Also, be sure think "energy and life" in your own seat when you want it out of him. It's hard to explain in words, but you add "life" to your seat, kind of inhaling and thinking "lets' go!!!"
The endless right/left, right/left of puttting the leg on will rapidly dull out many horses, if they have a natural lazy tendencey, and a two legged squeeze can make them resentful and balk back. a quick leg, leg , whip will be short, sweet and fair.