Where was the very first golf course in the US?

The first golf course in the USA was at Oakhurst WV:

1884: The Oakhurst Golf Club is founded at White Sulphur Springs, West Virginia. It hosted the Oakhurst Challenge, the oldest known golf tournament in the U.S. The Oakhurst Challenge Medal is recognized as the oldest known prize for golf in the U.S. The first hole at The Homestead survives from this course and is the oldest surviving golf hole in America.

https://en.wikipedia.org/wiki/Timeline_of_golf_history_%281851%E2%80%931945%29

Oakhurst Links is a Public, 9 hole golf course located in White Sulphur Springs, West Virginia. In 1878, Russell Montague moved to White Sulphur Springs, West Virginia, in part for its "healing springs" and purchased property a couple of miles from the Greenbrier Resort -then called the Old White Hotel. A few years later, Russell Montague offered his property as a site to build a golf course. The course was laid out and named "Oakhurst Links." By 1884, if not earlier, a golf club was formed consisting of Montague, Grant, Torrin, brothers Alexander and Roderick MacLeod and George Donaldson. Although often ignored in previous accounts of the early history of golf in the United States, Oakhurst Links predated the St. Andrews Golf Club of Yonkers by at least four years making it America's first organized golf club.

https://www.golfcourseranking.com/courses/west-virginia/white-sulphur-springs/usa/oakhurst-links/2571

Consider the Foxburg Golf Club (Pennsylvania), and The Oakhurst Golf Club (West Virginia). Both Foxburg and Oakhurst had records of playable courses in 1884. The Foxburg, however, was a private course on private property that only had eight holes when it was created. In 1887, the Foxburg added nine new public holes to their course, and that course has been in continuous working existence ever since. The Oakhurst Golf Club also formed in 1884. Oakhurst claims to be “the first organized golf club in the United States.” Remember, some golf clubs at that time were not exclusively golf clubs

https://www.golfwrx.com/472024/the-5-oldest-golf-courses-in-the-united-states/

America's oldest course put up for sale

As Keller tells it, Montague's doctor advised him in 1878 to move from Boston to a healthier climate. Montague chose White Sulphur Springs, partly because of stories about its so-called healing waters. Montague and a small group of colleagues built the course and held the first golf competition about 1884 in the Scottish match-play tradition, predating by a few years the St. Andrews Golf Club of Yonkers, N.Y.

But Montague and most of the original members eventually moved away. Play on the course stopped after 1910. The course was restored in 1991 when interest in old-time golf resumed. By modern standards, it's not much of a course: 2,235 yards; "longest hole is 356 yards and the shortest is 106." But it would have been tough with hickory-shafted clubs and gutta-percha balls, which is why the renovators required golfers playing it to use them.

http://sports.espn.go.com/golf/news/story?id=4401187

Based on the combined factors of founding date, the date when golf was first played, and continuous golf at the same location, it can be claimed that the oldest golf club in the U.S. is The Edgewood Club of Tivoli -- founded in 1884, golf in 1885.

The golf course at this club in upstate New York started with five holes in 1885, was reduced to three holes a short time later when some land was sold, and expanded to nine holes in 1916 when adjoining land was purchased. The course and the club's original clubhouse are still in use today.

As cited above for Vesper CC, several clubs were founded earlier but did not have golf until later years. Other clubs and courses (e.g. Oakhurst Links and the early South Carolina clubs) either have not had golf continuously or moved to a different location.