A definitive method of dicing an onion
Do I first cut it in half or do I make crosswise incisions before chopping it? And what is the best knife to use?
I use two different knives when I dice an onion. First, cut the onion in half, through the root end (so that each half has a bit of the root on it to hold it together). Next, take a paring knife (the skinny blade is why I use this knife here) and make vertical cuts that go almost all the way to the root. Next, use the paring knife to make horizontal cuts while using one hand to hold the onion in place. Last, use a chef's knife to make downward cuts to finish the dice.
If you are trying to julienne cut the onion, cut off both the top and root part of the onion and peel it. Then chop the onion in half vertically and cut thin slices of onion on an angle from the outside toward the middle (about a 10 degree angle should be fine). I do all of this with a chef's knife.
I'm sure there must be videos on youtube in regards to this, but here it is in a nutshell. Cut the ends off of the onion. Cut it in half from the root to the tip. You will have two halves now, each with half the root. On the end not containing the root, cut horizontally along the entire length up to the root, stopping just short of cutting all the way through. For a larger onion (or a finer dice), repeat this step as many times as necessary starting close to the cutting board working up to the top of the onion half. Then, cut vertically many times along the onion, also just to the root but not through. After this, start cutting at the leading edge (the side that does not contain the root). Cut all the way to the root, and you will have diced onion.
Also, the best knife to use for this is a very sharp french knife. Some people will also use a serrated knife, but there is no tip. The french knife is all you need.