TLDR: change your flight to BWI and take Amtrak.
Dulles is a bad choice for you.
Generally, people pick Dulles for one of two reasons: a) it's the "well-known, household name" DC airport... or b) it is actually the best choice for Charlottesville if you are renting a car and driving.
Travel agents who don't know Charlottesville can be forgiven for simply assuming you'll rent a car; the world's assumption about provincial America is that urban transit is awful to nonexistent. But Charlottesville is a college town, and they tend to have much better public transit than most.
Fly to National/Reagan/DCA. Take Metro to Amtrak.
Nate Elridge adds: Fly to DCA. Take the Metro 2 stops to Alexandria (it's on the same line). All the trains to C'ville go through there.
This wins for overall simplicity (only 2 trains and easy transfers: the Metro and Amtrak). The metro is in the airport; Alexandria Amtrak and Metro are right next to each other; and C'ville Amtrak is downtown and around stuff to do.
DCA is more of a commuter/short-haul airport, but if you can link it, it works great.
Fly to BWI. Take Amtrak.
Since you are flying internationally into an airport other than Dulles, it won't be too hard to get a flight to BWI instead of IAD.
It's barely a mile from BWI airport to BWI Amtrak. At this writing (mid COVID) the BWI to Charlottesville (CVS) fare is a measly $21.
There are several trains a day - regional service, the overnight Crescent, and the overnight Cardinal. Going towards Charlottesville, it does not matter which you pick. However, on the return trip from Charlottesville, pick the regional service, not the overnighters. By the time they depart CVS they will have already traveled for nearly 24 hours from Chicago or New Orleans, and can accumulate many stacked delays enroute. They can come in hours late.
There are only 3 trains a day, and some will require a transfer in Washington DC (WAS). But this is a GOOD thing. WAS is downtown, 2 blocks from the Capitol and the National Mall, Smithsonian, memorials, etc. There is a day-check for your bags for Amtrak passengers, specifically so you can do that! So I suggest designing your trip to have a long layover in WAS so you can do on-foot downtown tourism.
From BWI to WAS, there is very dense Amtrak and commuter service, rivaling European cities. So you can land anytime at BWI, and have only minutes of wait for a train to WAS. Learn where to go to get your CVS train, then day-check your bags and see the town. Then make your WAS-CVS train on time.
Fly to NYC or Philly. Take Amtrak.
All the above stuff, except fly internationally into JFK. It's a shlep to get to downtown NYC to get Amtrak, but they are at 34th and 7th - yes, 8 short blocks from Times Square. So you can do the same trick with layover tourism (but you would have to choose one town or the other).
Fly to Atlanta. Take Amtrak. (Slow but very direct)
One train that directly calls at Charlottesville is the overnight Crescent. That connects Washington to New Orleans via Atlanta. So you could "straight shot it" from Atlanta to CVS in one single vehicle, no fooling around with flights and commuter trains.
However, the Crescent being a long-haul overnight train, again has a vulnerability to delays stacking up over its long 24-hour run. That would not be a problem departing Charlottesville toward Atlanta, since the train only began its run. It could have delays overnight, so I'd pad the arrival time in Atlanta by 3 hours to cover any eventualities. Atlanta Amtrak is fairly remote and underserved by transit but once you are downtown, the Atlanta Metro can easily get you to the airport.
Fly to Chicago. Take Amtrak. (beast mode)
You'd have to time this right, because the overnight Cardinal does not run every day. But it calls at Charlottesville, so one-train ride. And it's a near 24-hour run, so at this point you are truly into Amtrak's "land cruise" scenery-tourism experience. However, the scenery is really something. It heads sharply south out of Chicago to get into the hills as quickly as possible, and it just keeps getting better and better, with the denouement being crossing the Blue Ridge mountains just before you step off at Charlottesville.
As usual with the overnighters, it is vulnerable to delays, so pack in a few hours of oops time. There are fun things to do near Charlottesville Amtrak if needed.
That's because America has a whole lot of highway crossings and very few grade separations, and the shared freight rail network is at overload on an ongoing basis. I've never seen a freight railroad put freight ahead of Amtrak. I've seen things that look like that to novices, but those were simply limitations of the track. (E.g. having Amtrak take a siding because the freight would not fit in the siding; it's the most expedient way to do that).