Hit has a base sense of violent contact, but is used figuratively to express any strong or emphatic contact.
The hit in
Head on over yonder and y'all'll hit the road sooner or later.
is reach or meet, just sexed up a little. It's worth noting that Americans will associate this expression with places that don't have paved roads (i.e., the sticks) and, sometimes, with the people who live there (yokels). It's a bog standard sense of the verb, though. (Sense II.11.ff. at the OED, which notes its odd history: the reach or meet sense of hit was the word's standard sense in Old Norse but mostly disappeared in English only to reappear in colloquial American speech.)
Similarly, to go or to set out on a journey was the base sense of strike, one of hit's primary synonyms. The hit in
Hit the road, bub.
has a long history into Old English as strike out upon and means step briskly, not so much that one needs to be running but that one moves sharply and emphatically away from this place.
It may feel a little odd even to a native speaker (notice the answers that only want to treat hit the road as a pat phrase) since modern English tends to use hit for blows from a fist or hand-held tool and uses kick for strikes from one's feet. It's part-and-parcel of the actual English term, though, back to Old Norse and Old English, revived by an American tendency to indulge in stronger and more violent means of expression.