The only railroad built primarily for passengers in Virginia, other than streetcar and modern commuter rail systems, was the Richmond, Fredericksburg, and Potomac (RF&P, now incorporated into the CSX railroad). It was designed to carry primarily people from north to south, using steamships on the Potomac River to connect from the northern terminus at Aquia Creek to Washington DC until 1872, when an unbroken set of tracks finally connected Alexandria to Fredericksburg.
Most railroads in Virginia were built to carry freight. They were mostly farm-to-market transportation coridors, prior to construction of the coal railroads into the Appalachian Plateau in the 1880's. Alexandria was a shipping port that incentivized farmers to trade in Alexandria by building the Orange and Alexandria, Manassas Gap, and Alexandria, Loudoun, and Hampshire railroads. Richmond-oriented investors built the Virginia Central and other lines to draw business to their port, particularly in competition with Norfolk.
When the Virginian Railway was constructed in 1909, it bypassed existing towns with passenger stations served by its competitor, the Norfolk and Western. Henry Huttleston Rogers did not want low-profit passenger trains to complicate the scheduling of the Virginian through-freights to Norfolk that were loaded with high-profit coal.
Competing railroads, when interchanging freight, built massive "yards" such as the Potomac Yard in Alexandria. When interchanging passengers, "Union Stations" were built.
In 1971, all but one railroad in Virginia abandoned the passenger rail market, and Amtrak began operations. The Southern Railroad tranferred its passenger service to Amtrak in 1979, unable to compete with air travel and interstate highways. Today, Amtrak runs long-distance passenger trains between Newport News, Washington DC (and points norh), Emporia (and points south), Danville (and points south), and Clifton Forge (and points west). Amtrak gets some commuter traffic on its routes in Northern Virginia. There is no long distance passenger rail service for Roanoke, and a trip between Richmond and Charlottesville will require a "mode shift" to buses or a long detour north to Alexandria.
Amtrak is a passenger rail system, and owns no track in Virginia. The Rail Passenger Service Act that created Amtrak guaranteed it access ("trackage") rights to use railroad lines owned by other railroads, to operate intercity passenger trains. Amtrak pays CSX and Norfolk Southern for the use of their rail lines. Scheduling passenger trains and negotiating costs is a major challenge for Amtrak officials, since the private rail companies see passenger trains as a headache that constrains scheduling of profit-generating freight trains.
Visions of expanding long-distance passenger service are reflected in various studies by the Virginia Department of Rail and Public Transportation, including a proposal for upgrading existing service between Newport News and Lynchburg to Washington DC in the "short term" (within the state's 6-year planning window for funding transportation improvements). In addition, the proposed Trans-Dominion Express, which could be completely independent of Amtrak, would connect Alexandria with Bristol, on the Tennessee border. In all scenarios, public funding would be used to upgrade the existing rail lines and acquire "trainsets" (locomotives and passenger cars).
A proposal to connect Dulles International Airport to Richmond with intercity rail services, including passenger rail, was floated in December, 2007. Consultant Jim Crupi advised Richmond-area leaders to build:1

1. Jim Crupi, Putting the Future Together, November 2007, http://lmrcommunity.files.wordpress.com/2007/11/putting_the_future_together_final.pdf (last checked February 28, 2008)