Vrftsjtryk

Next week will be my first time on the US and I'll need some help getting around.
I'll be arriving to Washington Dulles International Airport on the 11th of this month at around 3 pm and I need to get to Charlottesville before the evening of the next day. I've considered buying a train ticket from Union Station but I've been told that a Greyhound bus could be a better option. Should I buy a train ticket or wait until the next day and take a Greyhound bus in the morning?

PS: Will I be able to buy a train ticket at Union Station without booking it?

Fly

By far the simplest (but not cheapest) way to get from Dulles to Charlottesville is to fly. United Express operates several IAD-CHO nonstops, each under an hour on a regional jet. This flight, however, would cost a few hundred dollars when booked separately and on short notice.

edit It seems from the comments that you are flying into Atlanta and connecting to IAD, in which case it is worth at least looking into flying to CHO instead. It probably won't, because there will be a change fee and you may need to rebook at a higher fare, but it doesn't hurt to check. Charlottesville is not near any major cities and IAD, for a major hub, is very poor for intermodal connections.

Rideshare

There are groups where people can arrange rides like Hoo's driving? Hoo's riding? NOVA on Facebook, which caters primarily to University of Virginia students. UVa is a large university and there is a fairly steady stream of people going back and forth between the campus and the Washington area, although this will drop off dramatically once final exams end next week.

Ride times, meeting places, costs and so on must be negotiated separately, which may be tricky to do from overseas. Also, some drivers may be reluctant to pick up someone who is not a member of the university community. But in the ideal, you'd have a direct and relatively inexpensive ride and maybe some good conversation to welcome you to the country.

Train

Dulles, unfortunately, is not located near any passenger train stations. For heading to Charlottesville, which is to the southwest, your best bet would be to take a taxi to the Manassas station (MSS)— there is no good way to do this via public transportation— and take an Amtrak from there to Charlottesville (CVS). Services at the station are very limited and there are only three daily trains, so be wary of timing; you can buy tickets online and have the barcode emailed to you or delivered via their mobile app. The coach seats on the train almost never sell out except on holiday weekends, but buying tickets in advance may save you some money as well as peace of mind.

Washington Flyer has a monopoly on taxi service from the airport, and charges around $45–50 to Manassas (metered, not flat). You can try your luck with Uber or similar services for a cheaper fare.

If budget is more important than time, you will need to backtrack a bit and take public transportation from Dulles towards Washington as detailed elsewhere. When you reach the Rosslyn Metro station, transfer to the Blue Line to Franconia-Springfield and alight at King Street; the King Street station is adjacent to the Alexandria Amtrak station (ALX). All trains to Charlottesville stop in Alexandria after they depart from Washington.

Bus

There are no intercity bus stations near Dulles, either, and none of the bus options is especially convenient. Your best bet is probably to make your way to Washington (see directions linked above), where you can get Greyhound service from Union Station for under $40 if you book in advance.

If you can stay overnight in Washington, Megabus operates a much cheaper daily trip which departs in the morning. Similarly, you can also board Greyhound at Springfield (stay on the Blue Line to Franconia-Springfield until the end of the line), but you arrive too late in the day for either of the Springfield buses. There is a bus service called Home Ride which caters to UVa students and departs from Fair Oaks Mall in Fairfax County, but its schedule is extremely limited.

Rental car

To get around anywhere outside the largest cities in the U.S., you're almost always best off if you have a car, so consider renting a car. The drive from Dulles to Charlottesville takes about two and a half hours and would be almost entirely on well-maintained expressways.

Taxi/Airporter/Uber

There are a number of so-called "airport limousine" services which operate vans or shuttle buses from Dulles to Charlottesville. These have the advantage of direct and relatively quick service, but they can be expensive, and there is no easy way to compare prices or quality among the numerous options. From what I can tell, the lowest you can expect to pay is about $180 plus a tip for the driver.

By the same token, you can negotiate a fare with a taxi driver, or see if you can get an Uber or Lyft driver willing to go the distance, but I cannot venture a guess as to what that might cost.

The train is more comfortable, and in both cases you will have to get into Washington DC from the airport. They appear to be about the same time in transit.

However, there are only two or three trains per day. On Dec 11, there are departures at 4:50 pm (you are not likely to make this) and 6:30 pm. You will make the latter train if your flight is on time.

The cheap Amtrak tickets are not refundable and you may not be able to exchange it if you are in the middle of the air when a delay happens. There is a train on Dec 12 at 11:00 am. That train runs through to Chicago. Sleeping accommodations, which you do not need, sell out. I do not know if seats sell out. I doubt it, but as with planes there is an advance fare that is less that you can buy over the web.

You might also consider flying. There is a flight on the 11th, United Airlines, at 10:20 pm, only $143 at this writing, and no need to stay over. You might even be able to add this to your existing ticket.