Related: How to get to/from Washington Dulles International Airport?
Ground transportation at DCA continues to be severely impacted by construction projects.
Project Journey, a massive expansion and reconfiguration of the roads near the terminal, is underway until at least 2021. As a result, several exits at the B-C Terminal building are closed indefinitely, ride services like Lyft and Uber as well as hotel shuttles have been relocated upstairs to the departures level, and due to the exit and lane closures, there are massive backups at the taxi stand on the arrivals level.
Visitors should also note that the Yellow and Blue Lines of the Metro will terminate at the airport between May 25 and September 8, 2019, i.e. there will be no rail service to any of the six stations south of the airport. This does not directly affect passengers transferring to Dulles; however, it may lead to more severely crowded conditions at other chokepoints in the system.
[answer for "normal" operations]
Metro (rail and bus)
Outside of heavy maintenance windows where segments of lines are shut down, the most reliable way to get from Reagan National to Dulles is still transit, despite the well-publicized problems that have come to light since the 2009 accident; a taxi comes second, and other options distantly below that.
The simplest route, and the one the WMATA Trip Planner will always favor, is to take the Blue Line in the direction of Largo Town Center, exit at Rosslyn in Arlington, then transfer to Metrobus 5A, which will take you to the main terminal at Dulles with only one stop. Traffic delays are to be expected, but the schedule is somewhat padded to accommodate; I have never been on a 5A that was more than 10 minutes delayed in either direction.
The train requires a SmarTrip card. Total combined regular fare is $9.50 at this writing (maybe 20¢ cheaper off-peak). I would expect the entire trip from terminal-to-terminal to take around an hour and a half. Caution for the future visitor: there have been repeated attempts to cancel the 5A bus, and more are likely to come in the future, especially after the Silver Line extension to Ashburn opens (perhaps 2019 or 2020).
If you prefer a longer rail portion of the journey, which has less potential for traffic problems, you can transfer at Rosslyn to the Silver Line, taking it to the end of the line at Wiehle-Reston East. From Wiehle there is a shuttle bus from the parking garage located to the north side of the station, the Silver Line Express, which costs another $5 which is collected when you arrive at the airport. Note that the shuttle is operated by a company under contract with the airport authority, not Metro, and they do not accept SmarTrip for payment. If you can't wait 20 minutes for the next bus, you can usually find taxis waiting at the station.
As an alternative to the Blue Line, you can take a northbound Yellow Line train (toward Fort Totten, Mount Vernon Square, or Greenbelt) to L'Enfant Plaza and board the 5A bus from there. You are backtracking and adding bridge delays to your trip, but it may be easier to find a seat at L'Enfant, and the Yellow Line runs more often than the Blue. If you really want to kill time, you could also just transfer to the westbound Orange/Blue/Silver at L'Enfant and proceed from Rosslyn or Wiehle-Reston East as described above.
Taxi or car service
A taxi or car service is the next natural alternative, but of course subject to the vagaries of some of the worst traffic congestion in the United States. There is rarely, if ever, a traffic delay to be encountered on the Dulles Airport Access Road; the problem of course is getting to the entrance to this road in the first place, which requires time on I-66, the Dulles Toll Road, and/or VA-123 through Tysons Corner. Depending on the time of day I would allot forty minutes to ninety minutes.
The airport authority estimates DCA-IAD fare at around $70 not including tip; unlike some cities, there is no flat rate for taxi rides between airports in the Washington area.
SuperShuttle serves DCA and IAD, and there are other shared van services in the area. It should be cheaper than a taxi for a solo traveler or small groups; SuperShuttle quotes me $32 for the first shared ride adult passenger for a one-way ride from DCA to IAD, and $10 for each additional passenger. The trouble with shared vans is that you do not have much say in when you depart and how you are routed. The operator will want to delay departure to take as many people in the van as possible, and the driver may prefer to drop people off first in various parts of Northern Virginia, compounding your exposure to traffic delays.
SuperShuttle now offers a a non-stop ride option; it quotes me $62.
One-way car rental
At the time of day you are traveling, it wouldn't make sense to rent a car, but very early in the morning or very late at night, when transit is not operating, it can make sense in some cases (e.g. a group traveling with numerous pieces of luggage).
At this writing, ZipCar is not available at either airport. You can pick one up in Crystal City near DCA, but there is no drop-off point accessible to IAD, except at the transit stations indicated above.
As for checking bags, the airline would be the best authority on whether your bag could be checked through. But even in the heydey of the now-defunct UA/US partnership, with USAirways' focus city at DCA and United's hub and gateway at Dulles, a DCA-IAD transfer was uncommon. I really doubt you would be able to do so and have it arrive the same time you do at your final destination; I would plan to reclaim and recheck.