Wow - getting free wine advice from Mark Slater - most people pay $150 a head for the same! Of course, some extraneous food comes with it ... There's plenty of good advice here, actually, but I'd like to expand on one: store tastings. In DC, most stores open a few wines each Saturday for customers to sample. Some are purely marketing (look out for the glam wine girls at Magruders, usually pushing Red Bicyclette ...) Some, however, are actually instructional. Bells on M St b/w 18 and 19 opens 10 bottles each Saturday at 12:15, the tasting goes to 2 pm. Their selection is somewhat idiosyncratic, but the emphasis is on CHEAP and good. The tastings often include wines they don't have yet, but they want to get some customer reaction. Anyone is welcome, and the group stands around and trashes the wines as they themselves get trashed. Well, not really, no one's drinking that much, but you get the idea.
Their only rule is that you have to taste in order - as Bob Luskin, one of the owners, says, "If I suffer, you suffer." So you will have to taste a white or two to get to the reds, but as you do so, and listen and discuss the wines with everyone else there, you will learn more about what you like and dislike in each sample. You will hear a lot of, "That's a good $10 wine - too bad it costs $20!" Arrowine [mentioned by previous posters] is indeed another good store. Look for their tastings that feature a particular importer. Get to know the importer's name and style of wine. ("Read the back label" is one of my mottoes for learning about imported wines - if you like one Robert Kacher Rhone, you may like another.)
As you focus on inexpensive wines, don't ignore stores that feature costly ones. The Vineyard in McLean, or Wide World of Wines on Wisconsin in Glover Heights, are both known for higher-end, quality wines, but they also have some bargains. Remember, if the $10 bottle impressed the guy who mostly sells $50 wines, it is probably pretty good.
Find some wine writings you like. This site is a good source for information, especially on what's good and available in this market and where to find it. At the risk of, ahem, shameless self-promotion, I try to highlight high-value, affordable wines in my writing on my Web site, dmwineline.com. Not systematic writing, just fun and idiosyncratic, and hopefully helpful. And as you learn about wines, and which ones you like or don't, you may find yourself paying more attention to the world around you - at least in an olfactory sense. My other motto for learning about wine is "Smell everything, taste with discretion." You'll hear people talking about "barnyard" or "rosehips" or "cat's pee" in their wines. You won't be able to pick up the hint of violets in a good Rhone grenache if you don't enjoy your surroundings. And when you and your future wife have children, and your kid sticks her nose in your glass and says, "This smells like green beans and monkey farts," well, you'd better know at least half of what she's talking about. Some practical advice for $10 reds: Bonny Doon's Big House Red Argentina (Alamos, Trumpeter - Malbec, Cab, Merlot and Pinot) Chile (Cousino-Macul, Santa Rita 120, 2 Brothers - Cab, Merlot, Syrah) Cheers!