Find a Hotel Outside of City
Distances in Washington depend heavily on how far you are from the subway (also called the Metro). If you're in the thick of things but far from the subway, then you're going to spend a lot of time getting around. If you're out of the way but close to the subway, you can get to places quickly.
Find a List of Hotels
First, go to expedia.com to find hotels deals. Do a search for Washington DC and the dates you'd like your vacation package. When you get your results, scroll down and you'll see a map with Washington DC's different areas. Click on Arlington or Alexandria and then click on "Show Hotels Near Arlington"
Hotels in downtown Washington DC aren't cheap, but you should be able to find quality hotels in Arlington or Alexandria closer to $100 per night.
Check Distance Between Hotel and Subway
Get the hotel's address from Expedia and then go to wmata.com to see how far it is from the subway station. In the upper left hand corner you'll see a trip planner. In the "from" location, enter the hotel's address. In the "to" location, enter a generic spot downtown like "White House". Then click on "Rail Only" (I wouldn't recommend taking the bus on a vacation. The subway is easy but the bus system could get confusing).
Finally, click submit and you'll get your results. It should say "Walk __ miles from [hotel's address] to ____ metro station." Look for something less than a half a mile from the metro station.
Purchase Restaurant Gift Certificates
You can make eating a lot cheaper if you use restaurant.com to purchase gift certificates beforehand. At restaurant.com, you pay $10 for a $25 gift certificate. If you search for a coupon on slickdeals.net, you can make restaurant.com even cheaper.
For this step, repeat what you did for the hotels. Find the restaurant's address on restaurant.com, then check how far the restaurant is from the subway on wmata.com. Be sure to choose restaurants close to the subway, otherwise you might have trouble getting to them.
There are lots of Washington DC restaurants on the site to make dining cheaper.
Buy a Metro Farecard
Be sure to purchase a MetroRail unlimited pass. It'll cost $7.80 for a 1-day unlimited pass or $26.40 for a week long short trip pass (the short trip thing is confusing, but don't worry, you shouldn't be taking any long trips).
You can buy them online if you plan ahead (http://wmata.com/fares/purchase/passes.cfm) or just get them at the airport's subway stop.