How can I safely improve my cutting technique
All of the following play an important role in cutting technique/speed:


Practice! Probably the single most important.
A very sharp, clean knife. Always hone your knife before use, and have it sharpened regularly (6-18 months depending on use)
A fast and stable cutting surface. A solid end-grain cutting board is ideal.
The food should be stable. As Nick says, make a flat side to your food if it doesn't have one.
Proper off-hand placement. Your off-hand should be curled with your finger tips resting on the food. Your first knuckles should be against the side of the knife blade. Your fingertips should be tucked out of the way due to the curling. Your thumb pushes the food under the knife as you slice.
Proper knife-hand placement. This can vary based on your biomechanics, hand size, knife, etc. However, the gist is:

Don't hamfist it
Relax your hand and arm
Gripping high up on the bolster of the blade can help with control and speed. I pinch the blade itself between my thumb and forefinger.

Unless you're actually chopping (e.g. using the rocker technique to chop herbs quickly) you should actually be slicing your food. A slight forward movement of the knife blade as it passes downward through the food decreases resistance and speeds the cut. It also is gentler on your food.


To be completely safe you could use a finger guard. For example, Jamie Oliver Finger Guard. I've never used one of these, but it seems like it would significantly decrease the chances of a mishap. It also seems there are a few other brands out there to try.