I'll give this a stab (no pun intended
). If by "Cook's Knife" you mean a French Chef's knife, then it is a very handy knife indeed. It can be used for a number of tasks such as, but not limited to; slicing, dicing, chopping, carving, mincing, and chiffonade. The point can be used to extract the stem and woody portion of the tomato, or detail work while carving veggies. It's curved belly makes it ideal for slicing through both hard and soft veggies, using a sliding/rocking motion, while the heel of the blade is used for chopping.
For most uses, hold the knife with your thumb and curled forefinger pinching the blade spine just forward of the bolster (the part where the blade connects to the handle), and the heel of your hand resting on top of the handle and remaining fingers wrapping around the handle. This gives you control over the vertical angle of the knife while allowing you to move the knife up and down, forward and reverse, as well as controlling any side to side movement. To slice veggies such as celery, carrots, etc., lay the veggies on a suitable cutting board, with the long axis laying sideways to you. Grasp teh knife in your hand as described above, and place the point behind the veggies, with the cutting edge raised above the veggie. Push the blade onto and through the veggie while sliding the knife slightly forward. Raise and repeat. To control the thickness of the cut, grasp the veggie with the other hand, with the tips of your fingers holding the food, and your knuckles curved slightly forward. Use your knuckles as a guide against the side of the knife to deep the cutting edge away from your finger tips. Of course you don't want to raise the blade so high as to bring it down on your knuckles. Keep the blade side against your knuckles and your thumb tucked back, away from the cutting edge. It is natural to try to hold veggies between your finger tips and thumb. This will result in a sliced thumb. Most of us around here have experienced that pain.
Start practicing this technique slowly at first. Speed will build with time.
Chopping is done in a similar fasion, but without sliding the knife forward. Knives that chop well include the French Chef's knife, the Santoku, and the cleaver. The technique is the same for all of them. Again hold the knife as explained above. But instead of cutting with a sliding motion, simply raise the heel of the knife and press straight downward. Lift the heel, slide the food back under it and repeat. When you become skilled with the knife, you will learn to lift the entire blade and bring it back down onto the food. This can be a very fast way of chopping food, but can also shear off finger and thumb tips quite easily if you do this carelessly.
Paring kives are used for such tasks as peeling potatoes, apples, and other firm-fleshed veggies. They are also used for detail work when sculpting foods. Some people use them for slicing potatoes and carrots into a pot, but I have never been comfortable with that uses as it begs you to cut yourself. I use a cutting board rather than my thumb to cut against. The cutting board doesn't bleed or say ouch.
A serrated carving knife is probably a bread knife. Most carving knives have a smooth cutting edge to avoid tearing the meat, and maybe some grantons (depressions) carved in the side to help keep food from sticking to the blade. Knives with smoothly rounded scallops in the blade can be used for both carving and bread slicing fairly easily. But then again, I use a 10-inch Chef's knifed with a straight blade from the spine to the belly. I don't use hollow ground knives. This gives me great control weather I'm slicing very thin slices of cheese, veggies, or meat (either raw or cooked). So I don't own a carving knife, though I do own a bread knife.
For any of your knives to work, you must keep them very sharp. Thjis will make cutting almost effortless, and more efficient. This helps you avoid mistakes that cause slicing mishaps, such as slipping off of a tomato skin and carving your thumb instead.
There are a host of good knives out there, and I hear Global makes a good product. It's best if can can find a cutlery store and try a few brands, to see what really works for you. I love my Chroma knife, while I know of others who swear that Whusthoff-Trident is the only way to go. Still others love some of the more exotic Japanese knives or Shun knives made with Damascus Steel.
It all depends on what you are looking for. My daughter and I recently went to Pike's Place Market in Seattle and checked out a knife shop there. We found a very light, and super sharp knife made simalarily to a Global. It was exclusive to this shop (wish I could remember the name of the knife brand) and was light as a feather. The handle blended seemlesly into the blade, with no bolster. For her, this knife was a dream. She got to test drive it on some parsley. It cut through the sometimes tough parsely with no effort at all. The price for the 8-inch Santoku style knife she tested was somerwhere around $70. I wanted to purchase it for her on the spot. Unfortunately, my money went into the round-trip plane fair that got me to Seattle. Btu the point of this is; telt the knife. I like a slightly heavier knife than the oen we tested in Seattle, but not as heavy as the German knives. Only you can tell what knife you are comfortable with.
Seeeeeya; Goodweed of the North