My first investment in DC was 6 years ago when I bought my first house in an "Up and coming" neighborhood in the city limits. The house was in Petworth, and as those who live around here it has been one of the hottest markets for the past couple years. To the dismay of my Girlfriend, then Fiance, now Wife... the neighborhood wasn't the best. It did get better and better to the point that my the home price doubled (After some minor upgrades).
My advice: Start looking at some of the neighborhoods that have a lot of development and room for prices to grow. I really like the Fort Totten area (Anywhere near the metro) and Brookland. I would also consider homes within walking distance of the Anacostia metro station. I know Anacostia gets a bad reputation, but it is growing and developing and it also is one of the few places left in the city that you can still find some breathing room. Try to find a big home with the potential of a rentable inlaw suite, put some rehab into it and live there for a couple years. Believe me, it will pay off.
Just be sure to be security conscious. Get an alarm and really lock down all parts of your house. New homeowners in "up and coming" neighborhoods are big targets, but in my eyes its all worth it when you sell that house for double a few years later.
-Shayne