The dining fork is a relatively recent invention. Originally they were used primarily for the actual process of cooking or serving food rather than as a tool to place it in your mouth. It wasnt until around the Middle Ages that the dinner fork used today came around. Prior to its introduction people would simply eat with knives (the fork being more or less a more efficient version of a knife used to hold down meat while another knife cuts it) and their hands. As for chopsticks, it is assuredly older than the dinner fork, dating about 5000 years ago to the Shang dynasty. Its origin seems to be as a more refined version of using sticks to retrieve food boiled in large pots. Source- http://researcharchive.calacademy.org/research/anthropology/utensil/
/u/Occultus-'s Smithsonian magazine link is not really the whole story.
Yes, chopsticks did exist quite early in the history of cooking, but not in for eating. While they had quickly spread throughout East Asia, their use as eating utensils was more limited. The early use of chopsticks did correspond to cooking, but these too were different (and still are) from what you are using to eat – You cannot cook too well in a hot wok with short chopsticks. The most notable difference between cooking utensils and eating utensils, however, is in the cutlery. The Chinese vegetable cleaver (which looks like a cleaver you might see in a Western butcher shop but is much much thinner) is/was a sort of all purpose knife. There are broader and narrower varieties but for the most part it's an all purpose knife. But not one you'll ever see sitting on the dinner table when it's time to eat.
Even more recently they show up in neighbouring cultures but again not for eating — a common and in fact sole use for chopsticks in the area around the Burmese/India border was actually for preparing opium. This is still practiced today, but no one is eating with them.
Before the popularisation of chopsticks (and the fork, which is also a much more recent invention), people ate the old-fashioned way, with their hands (as is still the case in a number of cultures).
And actually even after the spread of chopsticks, they have not always won out in all domains. In Korea, for example, the traditional custom is that rice is eaten with a spoon, not with chopsticks. It has been suggested that this developed from the fact that rice often had other grains mixed in, meaning it was less likely to stick together and thus more difficult to eat with chopsticks.
If you're interested in food history, I highly recommend Consider the Fork: A History of How We Cook and Eat by Bee Wilson. It gets exactly to this sort of question about changing food habits. It's a good little read but covers a lot of bases and makes an excellent introduction into the history of modern and pre-modern eating.
Brace, C. L. (1986) Eggs on the face, ƒ in the mouth, and the overbite. American Anthropologist 88: 695–7
Wilson, Bee (2012) Consider the Fork: A History of How We Cook and Eat