I've cut my fingers dozens of times with cheap kitchen knives.
How many times do you think I've cut myself with a good chef's knife?
Once. An it was because I let the knife get too dull.
Cheap knives have flexible blades and are generally harder to cut with. You much more likely to slip and cut yourself if you don't have a proper knife.
I really trust a chef's knife to slice through the veggies. When I use cheap knives I spend a lot of energy just trying not slice my fingers open.
I know that chef knives are can be expensive. However, you'll use everyday, it will save you time, and you'll feel like a pro using it.
Convinced yet? Good, I have a few recommendations including a very affordable option.
Michael Chu, at Cooking for Engineers, has an extensive article of knives he has tested so I'll borrow 2 of his recommendations.
If you have small hands you might want a 6" knife instead because the larger the knife the more difficult it is to control.
MAC MTH-80 MAC Mighty Chef 8" with dimples $129 (This will be Happy Herbivore's next knife.)