Contrary to what may seem logical, use less leg. Don't nag the horse with your heels as this is usually what makes the horse dead to the leg in the first place. Start at a halt, ask gently with your seat and lower leg for some forward movement. If there's nothing, increase the pressure, bring in the crop and spurs if you need. As soon as they even take a step forward, release all the pressure and praise the horse. Keep doing this until the horse moves quickly off a small leg or seat aid between and in paces. Just use the basic training principle of pressure and release. Expect the horse to react quickly (read: be in front of the leg) to whatever aids you give him (as long as they are clear and precise), and expect the horse to keep going as you've asked them to until you ask him to stop or until he dies haha!
people will often talk about things like kicking,pushing,driving,squeezing releasing etc and this may usually help but from personal experience I'd have to say that the number one most important element to encouraging a horse forward is to pick up YOUR own energy,I mean if you aren't 100% positive on what you are asking your horse to do - why should he be - really being able harness and use your own energy is super important and surprisingly effective tool that is used by the majority of high level riders at home and in the ring - really get excited and show some determination (back it up with reasonable use of natural and or artificial aids) even if its a transition from walk to trot. You often see less experienced riders out and about kicking and whipping and carrying on all while they themselves don't even look ready for the transition and its simply because well, they aren't - you see it is so important to radiate confidence and enthusiasm to your horse because it not only instills motivation in the horse that he can do it but when you pick up your energy your body will naturally lighten and preform upward aids more precisely. I hope this helps a bit, lazy ponies can be a pain to start with and can surely be frustrating, but if you find yourself frustrated,out of breath and flustered try taking a brief walk on a loose rein,take a breath and try redirect that frustration into upward energy, your own energy is the first step to establishing a positive connection of riding between you and your horse.
I have noticed that when you have quick leg and make some transitions :)