The old adage, borne out of unflinching observation and repeated experience, is that the most dangerous thing in any kitchen is a dull knife. Think about extra pressure, the leverage against a stubborn piece of raw chicken or gristly steak being prepped for the grill, or even a simple raw tomato.
The blade doesn't slice smoothly, so that added pressure is exerted, the blade slips sideways and down, onto the cutter's finger. It happens every day. The answer, as is usual in these matters, is sublimely simple, cuttingly complex:
A sharp knife.
But all knives arrive in a kitchen with a factory fresh edge, so where can it go wrong, and where can it go right? It begins with self empowerment; in this case, with going to a store and asking to handle, or even test, the knives they want to sell. The actual cutting of veggies or meat is usually the domain of more specialized boutiques, and therein lies the rub. Price.
It is, under almost all circumstances, best to buy two, or perhaps three, knives, rather than a whole bushel of them, which usually come arranged in a somewhat attractive wooden block. A nice, but impractical, gift. But a paring knife, 2-3 inch blade, a standard blade of 6 inches or so and a big bad boy of 8-10 inches will absolutely be all you need.
The best of the best will cost upwards of $100 for each knife, so be aware of your needs. Chopping vegetables, even dicing such items as garlic or parsley (often both, for many dishes) requires that medium blade. The smaller blade is useful for coring tomatoes, getting rid of the green stems on peppers, and other more detailed work. The big blade is for carving, perhaps for gutting fish, and even for more volume in dicing. It's amazing what a great knife can do in a short amount of time.
There are technical considerations (forged, or stamped?), and the inevitable price point. But the best thing, really, is to simply pick out four or five knives, and handle them. Don't worry exactly about the cost, but hold the knife, feel if it tips heavy to the tip of the blade, or has too much heft in the handle. Get that hand feel you like, and then choose according to budget, based on what feels good in your hand at a price you can afford.
Brand name is not exclusively reliable, though, like tires or batteries, it is usually hard to really go wrong with a reputable brand. Wusthof, Henckels, and even the notably less pricey Victorinox, are fine, as are names such as MAC and Global.
Bottom line: test before buying, let your hand tell you if the instrument is right for you and keep it sharp. No more needless blunt force trauma, just thin sliced carrots and celery, French cut green beans. The only blood should be gently seeping from that perfectly pink inner core of Sunday pot roast.