Also are you going to cut the vegetables in julienne strips, dice, slice or cube them. Also important to how you want the end result to be.
There is a right way and a wrong way to hold a knife while you cut anything. Don't just grip the handle but hold the blade right above the handle. This will keep the knife from spinning and give you more control of the knife. Now your other hand is most important, it will control the size of the vegetable. Curl your fingers inward so your knuckles are the outer most part of your hand. Grip the vegetables with your left hand and the knife with your right hand. Keeping the tip of the knife on the cutting board, raise the back of the knife where you are gripping it and let the blade feel your knuckles, not in a way to cut them of course, just resting next to the side of the blade, push the knife up and down in a rocking motion, moving your fingers back each time you go upward. Be consistent with your size. It will feel awkward at first but you will get the idea after a few tries.