Integral calculus, which measures the area of various shapes that don't have convenient, straight-line boundaries, began long before the birth of Christ. Differential calculus, which deals with the instantaneous rate of change of a varying quantity, did not arise until the 17th century. Still, we present differential calculus first, because it is easier to understand. Then we develop integral calculus, and finally we show how they are related to each other. Most text books take this approach - but not all.
Calculus rests on a foundation of limits and sequences. Some people never worry about this foundation, and you can definitely get to the moon without it, but if you want to be rigorous, you should review the section on limits first.
Both Newton (biography) and Leibniz (biography) developed calculus independently, using slightly different notation, and argued over who deserved the credit for years after the fact.