3rd St. is labeled as a bike route, but don't do it. At least, not west of Wilton. Traffic isn't so much of a problem - there is actually plenty of room for you, your bike, and the plenty of stuff you'll pile onto your bike after visiting the various stores (such as Trader Joe's) on 3rd.
The unfortunate problem with 3rd St. is that, while the main part of the road itself is kept up quite nicely, the shoulders are very poorly maintained. Riding on a 3rd St. shoulder definitely requires a mountain bike.