How should a family of five get from DCA American Airlines Terminal to Crystal City Station

Take the Metro at the airport, either the Yellow line towards Mount Vernon or the Blue line towards Largo Town Center. The trains come every 10 minutes; Crystal City is the next stop, several minutes away.

The Virginia Railway Express commuter rail runs on weekdays (not on weekends or Federal holidays) and from Crystal City, Fredericksburg Southbound has afternoon departures at 1:10, 3:25, 3:40, 4:25, 4:55, 5:30, 6:15 and 6:55. It takes about an hour and a half to get from there to Fredicksburg.

Crystal City is literally the next stop on the Blue or Yellow Line from National Airport.

For clarity, there is no special procedure to riding Metro. You have to buy a fare card, enter-ride-exit.

Is there a specific scenario we should be aware of?

Breaking the question into two parts:

How to get from the DMV to Fredericksburg, VA

There are two options:

1. The Virgina Railway Express(VRE) This option is cheaper, stops at Crystal City (which is closer to DCA) and the price isn't affected by earlier purchase decisions. The downside is that it only operates on non-holiday weekdays, has a relatively early last train and is only really feasible if you are travelling when a commuting worker would be travelling.
2. Amtrak More expensive, stops at either Union station in DC or in Alexandria which are slightly further from DCA and the best fares require advanced purchase. However, it offers travel during a larger range of times, better on-train amenities and faster service.

How to get from DCA to the train station (Crystal City or Alexandria)

1. WMATA Metrorail (the subway) DCA has a stop for the blue and yellow lines next to the airport. If you travel north, both lines take you to Crystal City. If you travel south, both lines will take you to King St-Old Town which is next to the Alexandria train station. Up to two children under age five can travel with a paying adult for free however there is no reduced fare for children five and over. The fare from the airport to Crystal city is 1.75/2.15$ (non-peak/peak) and to King St-Old town is 1.90/2.30$ (non-peak/peak). (Rate is prior to a June 25, 2017 fare increase.) Furthermore each paying customer requires their own SmarTrip card which costs 2$ and can be purchased at any station. It is also worth noting that DCA's Terminal A is a 10 minute walk from the metro station. However, American Airlines operates out of Terminal B/C so this is less of an issue.
2. Taxi/Lyft/Uber, etc. Readily available at the airport. Considering the short distance being traveled combined with the the fact that a large number of individuals are travelling who might require SmarTrip cards this option might be more economical.
3. Walk - Crystal City is only a little more than a mile from DCA and both locations are connected to the Mt Vernon trail so walking is feasible.