Depart, begin a journey, as in The ship was heading out to sea, or When do you head out again?
head out after. Follow or pursue, as in Since they knew the way, we headed out after them, or A police car headed out after the car thieves.
The American Heritage® Idioms Dictionary Copyright © 2002, 2001, 1995 by Houghton Mifflin Harcourt Publishing Company. Published by Houghton Mifflin Harcourt Publishing Company.