Holding the handsticks
Hold the handsticks a third of the way from the handle, or two thirds from the tip where the string is tied. Holding the handsticks nearer the centre gives you a much more solid & controllable grip. I wittered on about the reasons why in more detail in my Single Sticking (devilstick) tutorial.
The first roll
Start holding one handstick in each hand with the diabolo on the string & on the floor in front of your feet so that your toes are pointing into one of the cups. Make one sideways step to your subordinate side leaving the diabolo where it is.
Gently pull both handsticks to your subordinate side causing the diabolo to roll along the floor. If you pull too fast the diabolo will slide & bobble around. Start off very slowly then gradually accelerate. All you have to do is get the diabolo to roll; you don't have to break any speed records. As the diabolo moves past your feet lift the handsticks to pick the diabolo up off of the floor. You should be standing with both handsticks level & the diabolo slowly spinning in front of your shins.
Keeping it spinning
If you don't do anything the diabolo will gradually stop spinning & then get tangled up. Spin is one of the most important things in diaboloing. The faster a diabolo spins, the more stable it becomes & the more tricks you can do.
To keep the spin going whip the diabolo by repeatedly flicking your dominant handstick upwards while keeping your subordinate stick still. Flicking with the subordinate hand cancels out the work done by the dominant hand.
A common mistake is to use the entire arm to create a huge whipping motion. While this can accelerate the diabolo more you are more likely to either jerk the diabolo off of the string or cause the string to loop around the diabolo's axle as you bring your arm back down faster than the string can fall. Try & keep both hands level as much as you can. You only need to flick using your wrist, point the handstick down, then quickly snap upwards, your hand doesn't have to move position at all. The motion should feel a little like gently tapping a drumstick on a drum.
Stopping it tilting
Once you've mastered getting started & keeping the spin going, your next problem will be that the diabolo starts to tilt towards or away from you. It will gradually tilt more & more until one of the cups starts to rub against the string & eventually you get in a tangle.
To correct this you simply have to move your subordinate handstick forwards or backwards to match the way the diabolo is tilting.
If the diabolo is tilting away from you (the cup furthest away from you dips lower than the cup closest to you), then push your subordinate handstick forward.
If the diabolo is tilting towards you (the bell closest to you dips lower than the bell furthest from you), then pull your subordinate handstick back.
Keep moving your subordinate stick back & forth as necessary, the aim is to keep the diabolo parallel with the ground.
The most important thing to note here is YOUR SUBORDINATE HANDSTICK IS NOT THE ONE YOU WHIP THE DIABOLO WITH! Forgetting which hand is which seems to be a common problem when learning the diabolo. If you use the wrong handstick to correct a tilting diabolo you will just make it tilt even further.
The further you push or pull your subordinate handstick forwards or backwards the more effect you will have on the tilting diabolo, with practise you will be able to keep the diabolo level with only the smallest movements.
Turning round
Once you've got the hang of correcting a tilting diabolo you will be able to spin the diabolo for longer periods of time. If you are practising with an incorrectly weighted diabolo, or you find your self having to correct some severe tilting where the string rubs against one of the cups you will probably find that the diabolo starts to turn to the left or right. When you're practising on your own this isn't really a problem; simply turn your body round with the diabolo so that you are always facing the closest cup.
If you are showing off to an audience though you'll want to turn the diabolo yourself so that it is always facing the right way. This can be done by gently pressing a handstick against one of the cups. When trying to turn the diabolo be ready to do some quick tilt correction.
Firstly lift your subordinate handstick up high so that your dominant handstick is closer to the diabolo. Then point your dominant handstick inwards so that the stick is pointing towards the axle.
To turn the diabolo to your dominant side move your dominant handstick further inwards in front of the string. Press the side of the dominant handstick against the top of the furthest cup. To turn it to your subordinate side move your dominant handstick behind the string & press it against the top of the closest cup. Use the side of the stick rather than the tip & remember to press gently, too much pressure will cause the diabolo to slow down & stop spinning.
That's all there is to it. Once you can control a spinning diabolo the next thing you will want to learn is How to throw & catch a diabolo.