There’s also the free shuttle Kennedy Center shuttle that leaves from the Foggy Bottom Metro every 15 minutes. When you take the shuttle back there’s a different shuttle that will take you to directly to Georgetown for $1.
Not near downtown - but I don’t think you’re really interested in downtown (museums & business district) anyway. I think you’re talking about midtown (Dupont - Georgetown) areas. Check out this map. Georgetown is to the left of Washington Circle (about a half mile northwest of the KC just off the edge of the map). Note - the map is rotated 90° counterclockwise (numbered streets run north-south). Virginia Ave points in the direction of Georgetown.
Kennedy Center has a shuttle that is, I think, free. I don’t know for sure because it’s only a 4 block walk from the Foggy Bottom station and I have always walked.