What is the date that pennies begin to be worth more than one penny?

There's no specific date because it depends on the coins' conditions and mint marks. However, most common date wheat cents from the early 1950s are worth at least 2 or 3 cents if they don't show too much wear, so that's a ballpark place to start.

If you're interested in finding cents that are worth more than a dollar, for example, in general you have to work your way back to the early 1930s and even some of those are common enough to be available at pretty low prices.