If you have a microwave, here is an easy way of preparing onions for cutting. Trim off the ends of the onions - don't peel them. Place them on kitchen paper in the bottom of the oven and microwave on High for a minute. You will be able to remove the skin more easily and won't cry when you chop them up.
Alternatively, leave the root end and cut off the top end only. Then peel back the skin over the root end and hold onto it as you cut the onion. The milky juice at the root end is thought to be the cause of tears.
If you want to chop and dice an onion finely, cut the onion in half and place one half cut side down on the chopping board. Hold the onion with your fingers. Make a series of horizontal cuts towards the root but not through it. Then cut it vertically from the top end towards but not through the root. Finally, cut the onion diagonally. Continue chopping for finer pieces.