Since half of the earth is day and half is night, what if I go to where the day and night meet?”

That place is called the Earth’s Terminator. It is a permanent band encircling the earth creating a twilight zone that travels around the earth at 1,036 miles per hour. (measured at the equator)

One half of the terminator precedes the dawn while the other half precedes night fall.

You don’t have to go to the terminator it passes over you twice each day.