How can a railroad be "Transcontinental" when it starts in Omaha, which is 1000+ miles from the nearest ocean? It depends on what you mean by "Transcontinental Railroad". Back then, when people talked about a "Transcontinental Railroad", they meant the entire (already existing) railroad network becoming "Transcontinental"; not a particular railroad company being a "Transcontinental Railroad".
Prior to the building of the first railroad line to California there was a highly developed railroad system between the Mississippi River and Atlantic Ocean. This network consisted of hundreds of privately owned railroad companies totaling about 30,000 route miles in 1860. However, there was no real commercial need for railroads west of (roughly) the Missouri River as there was little potential traffic; which means no profits; which means no investors. And, the Federal Government didn't build or fund infrastructure then. That all changed as the Civil War loomed. Both sides thought that a railroad to California would be of great value to "their side". It turns out it probably didn't matter all that much as to who eventually won the war, but "The North" was the side which was able to fund what eventually became the first link between California and the rest of the nation (completed by two separate railroad companies in 1869). And, this link may have kept California from becoming its own country or being reclaimed by Mexico. This same scenario played out in Canada too, when British Columbia hinted that they'd be better off joining the United States. This caused the Canadian Government to help fund the Canadian Pacific Railroad, which became the first Atlantic to Pacific railroad company (as a single company), making it the first "true" Transcontinental Railroad (completed in 1885).
Oh, unless you consider the Gulf of Mexico to be part of the Atlantic Ocean. If you do; then the first "true TRUE" Transcontinental Railroad (as a single company) was the Southern Pacific which created a route between Los Angeles and New Orleans (completed in 1880). But, some people say that doesn't count because the Gulf of Mexico really isn't really thee Ocean.
But, there was a railroad across the Isthmus of Panama, before any of the others, which - of course - connects the Atlantic to the Pacific. But, everybody says that shouldn't count because it's not a continent (but, Omaha isn't on the Ocean either).
The button below is an excellent site for all things related to the US Transcontinental Railroad companies, and specifically the Central Pacific Railroad.