I used to do a similar commute (Takoma DC to Farragut North) and took the first route SilverSpring suggests (Piney Branch to 13th to Colorado to 14th). The only downside is that the bus routes go down 14th so you are always interacting with buses and you specified that you were looking for something a little quieter.
Instead of a quieter route, I would suggest looking at the WABA website and taking a Confident City Cycling course. This would give you more practice on your road bike and also give you tips for riding on the busier city streets. That way you can take the easiest and most direct route. It is not hard to get used to with a little practice.
Good luck!
Liz