head for the hills vs go forth

head for the hills and go forth both are verbs.

Word	Noun	Adjective	Verb	Adverb
head for the hills	No	No	Yes	No
go forth	No	No	Yes	No

As verbs, go forth is a hypernym of head for the hills; that is, go forth is a word with a broader meaning than head for the hills:

Other hypernyms of head for the hills include go away, leave.

head for the hills (verb)	go forth (verb)
flee; take to one's heels; cut and run	come out of
	go away from a place