Based on the Rehoboth Beach Transportation website, it seems there are a number of options; the site does mention buses, including charters, but it does not explain every place those buses go. I enclose the link to the website so that you can contact them directly.
To get from Grand Central Station to Wilmington, Delaware, you can take an Amtrak train directly from Penn Station, which is a short subway ride from Grand Central. Check the Amtrak schedule for trains heading to Wilmington, as they run frequently. Alternatively, you can drive or take a bus, but the train is typically the fastest and most convenient option.
Atlantic City, NJ if you take the US 9 toll ferry. From there you can take the NJ Transit Service to Philadelphia to connect to many other trains.
You can't. The only train station located anywhere close to Delmarva is the Wilmington DE Amtrak station. You could get on the Amtrak in Newark or Harrison and travel south to Wilmington, but from there you would have to drive. You go down DE-1 towards Dover and keep on that all the way to the state line, and from there you would be in Ocean City Md.
Dwight - Amtrak station - was created in 1891.
Danville - Amtrak station - was created in 1899.
Norman - Amtrak station - was created in 1909.
Longview - Amtrak station - was created in 1940.
Holland - Amtrak station - was created in 1926.
Gilman - Amtrak station - was created in 1986.
Connellsville - Amtrak station - was created in 2011.
Tempe - Amtrak station - ended in 1996.
Tempe - Amtrak station - was created in 1924.