I have heard various terms to describe one's travels East. Some people say, "I am going out East." Others say, "I'm going back East." Still others say, "I'm going down East." Which is correct?
In American English, it is conventional to refer to broad geographic directions of the country as up north, down south, back east, and out west. It is by no means required to use these pairings, if context explains why someone is moving back south (e.g. returning to your childhood home in Alabama) or up west (e.g. from eastern Colorado to the Front Range). Moreover, these should be understood more as directions and not necessarily regions; one can relocate up north from New Orleans to Chicago, even though Chicago is geographically in the Midwest, and there are many cities farther north.
The up and down for north and south are easily explained by their traditional positions on maps. The notion of back east and out west reflect the pattern of Anglophone settlement, which began on the Atlantic seaboard.
The sense of relocating out to some place is of venturing forth from the norm; consider any of the first three senses AHD gives:
- In a direction away from the inside
- Away from the center or middle
- a. Away from a usual place
On the other hand, moving back is a return to some place:
- In, to, or toward a former location
- In, to, or toward a former condition
The usual pattern from the colonial era through to the mid-20th century was for people in the eastern part of the country to move west; the mean center of population of the United States has moved west with every single decennial census, so it is natural that moving back east and moving out west would be more common patterns than moving back west or moving out east, as you can see in this NGram of AmE from the closing of the frontier to the present:
Down is a separate matter, as it has the conventional modern sense of somewhere lower (hence downtown, referring to the direction towards the southern end of Manhattan Island) or somewhere away (hence downcountry and down to the shore, and in Baltimore, down the ocean), but also some idiomatic uses that make sense if you consider the destinations to be some place away from a "center," especially the Mid-Atlantic*.
down east (U.S.): into or in the eastern sea-coast districts of New England, esp. Maine. Also as adj.n. Hence down-easter n. Also transf.
down south adv. into or in the south; in U.S. down the Mississippi; into or in the Southern States.
Now, there are a number of alternative explanations given for the origins of down east to refer to Maine and the Canadian maritimes. The popular theory is that it refers to areas that were downwind from Boston and other ports, and since ships sailed with the wind, there was no need for any up west counterpart. But down east also corresponds to moving away from the ports to the east, and given the geography of Cape Cod, it is not possible to sail in any other direction. Wikipedia posits a relationship to measures of longitude, but it is uncited, and I think this explanation spurious.
* By Mid-Atlantic I mean in the historical sense of the region between New York and Philadelphia, not the modern sense of the region around Washington, D.C. The designation of an area rather north as Mid-, and the shift of that term over the last few decades to include Virginia and even North Carolina, is I think a neat illustration of where America's "center" has been shifting.
Depending on what you intend to express, then, back east, out east, and down east are all perfectly acceptable, if not always interchangeable.