When did people first start exploring space?

If you mean by observation, pretty much since man could draw. There are detailed "maps" of star patterns in languages too old for us to read, and there are pre-Judeo-Christian models of the night sky. Moon gods are a common religious theme - pretty much forever.

If you mean physically, NASA's space program was initiated during Kennedy's administration in response to the Soviets. Why? We wanted to be there first, and technically we were (at least, on the moon.)