Try to get as close to rock creek park as possible -- check out this place I just found, after being curious about your question. http://www.trulia.com/property/45347293-3024-Military-Rd-NW-
I do not think you should feel limited to the suburbs with your budget. If being near public transportation is your priority, be prepared to sacrifice renovations and property size for access, but I think it is definitely worth it. Also, consider getting a property with a basement/apartment to rent out to subsidize any extra $$ you may want to spend to get a better location. Good luck!!