Want to drive a chef crazy? Hand them a dull knife to use. For them it is equivalent to taking away their car and giving them a horse and buggy, taking away their smartphone in exchange for a flip phone. Sure we have survived before with less, but once the primitive mind has experienced the enlightenment of a new tool it will go into withdrawals when said tool is removed. I don't want to rub two sticks together - give me my damn lighter!
A sharp knife is also significantly safer than a dull knife as it allows you to exert less force and will be more controllable - as an added bonus the clean cut will make it easier for the doctor to stitch your hand back together.
There are really only three knives you need to accomplish any task in the kitchen: chef's, serrated, and paring.
Chef's Knife
The primary cutlery tool in almost every kitchen. I wouldn't recommend getting anything smaller than 8 inches. Size matters - a larger knife will give you a larger lever for better chopping, smoother cuts in large meats, and, with practice, will still be percise enough for dicing a single clove of garlic. The larger the knife the more material and the more you will be able to sharpen it (although you probably will never be at risk of lack of material). You don't want to invest in a smaller chef's knife and then have length envy once you master a stubby little knife.
There are two primary styles of chef's knives: Western (French & German) and Asian (Japanese). Of course this is an over simplification and there are lots of knife styles but I will refer you to wikipedia if you want to read up on it. Western knives tend to have a larger angle on the cutting edge which makes the blade last longer between sharpenings and more resilient to hitting bones. Japanese style chef knives have a much smaller angle on the cutting edge which makes them significantly sharper. While the edge is more susceptible to damage and you don't want to chop bones in half with them, more chef's have been choosing them over traditional western knives. I chose my chef's knife after reading Anthony Bourdain's Kitchen Confidential (which I highly recommend):
"Most of the professional's I know have for years been retiring their Wusthofs and replacing them with the lightweight, easy-to-sharpen and relatively inexpensive vanadium steel Global knives, a very good Japanese product which has - in addition to its many other fine qualities - the added attraction of looking really cool."
I have the 8 inch version and the blade is still in great shape after continued use for 3 years, of which, I have shamefully still not taken the time to sharpen it. It is still in good shape but I can no longer shave with it, making meals far less entertaining but maybe more sanitary. Moreover, I think it is an awesome value and you can regularly find them on sale. Although some days I wish I would have gone with the 10 inch...
Serrated Knife
Serrated knives aren't as versatile as a chef's knife and are a royal pain in the ass to sharpen, but you need one for two reasons: bread and tomatoes. The sharpest knife I have has an extremely smooth and polished edge. The lack of micro serrations make it glide against the smooth skin of tomatoes instead of biting into them - that's right the sharpest knife I own sucks at cutting tomatoes.
First and foremost, don't invest in an expensive serrated knife. It is a waste of money as it is annoying to sharpen and is really only going to be used for bread and tomatoes. Pick up a cheap serrated knife and if you mess it up while sharpening just buy a new one. I have another Anthony Bourdain recommended knife (can you see a pattern?) that has also been going strong for 3 years.. I think I originally got it for $20 so it looks like the price has gone up since then but don't worry, basically any serrated knife should do and I wouldn't be too sad if I have to get a different brand.
Paring Knife
At this point you have everything required to dice, chop, and slice your way to a great meal. Do you need a paring knife? No, almost everything you can do with a paring knife you can do with a chefs knife. Sure it may be a little awkward to peel a potato with a 10 inch knife, but it can be done. However, I love my paring knife and I use it religiously. Whether it's opening an avocado, using the thin blade to glide through something sticky, or coring an apple - I will reach for my paring knife. Plus, sometimes you just don't feel like breaking out the big guns for a piece of garlic.
My knife of choice is the 3.5in(9cm) Wüsthof classic Ikon. I am in love with this knife. Its handle is smooth without any harsh corners and feels divine in your hand regardless of which way the cutting edge is facing. Swing by one of your local kitchen stores and hold one in your hand, you will be sold. Moreover, the tip of the blade comes to a crazy sharp spear point which is perfect for round cuts, detailed carving, or jamming into a watermelon for stress relief.
Knife Care
I am getting a little tired of typing at the moment so if there is interest in a more detailed explanation just let me know. Here are the basics:
Don't put your knives in the dishwasher. Knives are typically heat treated and the high temperatures in a dishwasher can actually make the metal softer and your blade more crappy. Also the harsh detergent can degrade the handle especially if it is made of wood. I don't worry too much about the serrated blade but if someone puts my other knives in the dishwasher I get sad.
Use a wood cutting board and don't smack the blades on hard things (plastic cutting boards are fine too but wood is the way to go). Basic material science lesson: whenever two things rub against each other the softer material gives way first. If you use a glass cutting board, cut on a granite counter top, or generally smack your blade against other ceramics or metals you will damage the blade. Don't throw your knife into a sink full of dishes. Your knife should be the first thing you clean and put away before enjoying your meal.
Wash with warm soapy water and dry with a towel before storing in a safe place where the blade won't bang against other metals.