(Part of The Cookbook)
An important revelation for me was that every knife cut you make in the prep phase affects the cooking phase. How could it be otherwise? The smaller the item you're cooking, the less time it takes to cook. Not just that, but the shape you cut things into affects their cooking, too. A carrot that's cut into long and flat planks will cook faster than the same-size carrot cut into cubes, because there's more surface area in the first shape to absorb heat. This is probably obvious to you, but I'm a bonehead sometimes.
So, by paying attention to the size of things I was cutting, I became a better cook. If you're cutting carrots, try to get them the same size and shape, and they will be ready at the same time. If you're cooking multiple things together (like you usually will), cut them so they'll be ready at the same time.
Cook this recipe a few times and you'll understand what I mean really well: Red Cabbage & Cashew Slaw, Stuffed Collard Greens and Red Quinoa with Caramelized Fennel
It's important to hold a knife correctly, but what you do with your other hand (your "guiding hand") is even more important. You must learn to use "the claw." You hold the item you're cutting, but you tuck your fingertips under your knuckles and you tuck your thumb behind your fingers. This feels really awkward at first, but it pays off, because if you keep your guiding hand in this position, it's nearly impossible to cut yourself. Eventually, you will develop a sense of your thumb or fingertips being in danger if you slip out of the claw position.
If you want lots of practice cutting things, buy a bag of potatoes. Practice your hand position and also practice dicing at the same time: Cut a flat bottom and rest the potato on it, then cut 1/2 inch wide planks, take the planks and cut them into 1/2 inch wide batons, then take the batons and cut them into 1/2 inch wide cubes. There, you just medium-diced a potato (small-dice is 1/4 inch on a side, and large-dice is 3/4 inch on a side). Repeat until you're out of potatoes, then make a bunch of mashed potatoes.
I hope you like mashed potatoes!