Related: How to get to/from Washington Dulles International Airport?
[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, as you do not need the WMATA Trip Planner to tell you, is to take the Blue Line from DCA in the direction of Downtown Largo (northbound), then transfer to the Silver Line at Rosslyn in the direction of Ashburn (westbound, on the lower platform).
The train requires a SmarTrip card which can be purchased at the station. The total combined regular fare is currently $3.85 off-peak and $6.00 during peak hours. In the best case scenario this trip will take about an hour and 15 minutes, but may take up to 20 minutes longer due to headways or delays, especially during off-peak hours.
If you have a lot of time to kill and prefer a longer train ride, you can alternatively take the northbound Yellow Line (toward Fort Totten, Mount Vernon Square, or Greenbelt) and transfer to the Silver Line at L'Enfant Plaza. Note that with the opening of the Silver Line extension, Metrobus 5A bus and the Silver Line Express shuttle bus have both been discontinued. [Note: In September 2022, Yellow Line was suspended for rehabilitation of the bridge over the Potomac River. Estimated return to service is May 2023, but timely completion of Metro projects is unusual.]
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 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 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. The trouble with shared vans is that you may 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.