Here are some basic professional techniques for chopping vegetables and herbs.
Always use a sharp chef's knife, and tuck under the fingertips holding the food to keep them from getting nicked. For julienne or very small dice, begin by slicing a vegetable very thinly lengthwise. To make julienne, stack several thin slices on top of each other, and slice them into matchsticks; to dice, gather the matchsticks together and chop them into equal pieces.
"Chiffonade" refers to very thin strips of lettuce or herbs, like these basil leaves. Stack several leaves, with the largest on the bottom. Roll them up, and thinly slice them from one end of the roll to the other.
The best way to chop an onion is to start by cutting it in half from top to bottom; then place the cut halves, flat side down, on a work surface, and slice off the stem ends. Remove the skin, and make vertical cuts lengthwise without cutting through the root end, which holds the onion together as you work. Make a few horizontal cuts from the cut edge toward the root end, then chop across the onion to make cubes.