Davis Square to Horn Pond in Woburn. The whole west side of the pond is an old parkway that has been closed to car traffic. You've got the pond on one side and woods on the other.
Davis Square to Lowell, the first half of which uses the Minuteman Bikeway and Narrow-Gauge Bikeway in Bedford. This is a good way to get to the Lowell Folk Festival, July 27-29.
Davis Square to Andover. Almost all of this is on quiet back roads. This route, plus a short jaunt north on Route 28, is a good way to get to the Bread & Roses Festival in Lawrence on Labor Day, September 3.
Davis Square to Concord, via the Minuteman Bikeway and the stone-dust Battle Road path through Minute Man National Historical Park in Lexington and Lincoln. This is neither the shortest nor the fastest way to travel, but it travels through pleasant historic scenery. You'll appreciate the shade on a hot summer day. The section of Mass. Ave. in Lexington, between the two off-road trails, has a substantial hill climb in both directions.
Davis Square to Winthrop Beach, also known as the Scaryville ride, through the heart of industrial Everett and Chelsea. Very different from the rest of these routes! I don't recommend this for narrow-tired road bikes.