Along the old Berlin wall
The Mauer-Radweg follows the former wall's entire winding route - all 174 kilometres of it. You can cycle the entire track in one go, or you can decide to stop overnight along the way. The trail is signposted and divided into sections: you can reach each starting point by train or underground. If you are short-term visitor, maybe it's wise to opt for a halfday tour; many guides offer them.
Around the Grunewald (Green forest) in the west
Take the train up to the S-Bahnhof Grunewald, and look for the signs. Your pleasant bike ride through the shadowy forest will also take you to the Teufelssee (the lake of the devil) and the historical landmark of the Grunewald Turm (tower). If you're brave enough to climb it, you'll be rewarded with a splendid view.
Around Müggelsee for some vintage GDR
The big and the small Müggelsee in the East of Berlin are a perfect destination for an easy bike trip. Take the S-Bahn (take your bike with you on the train or rent one when you're there) to Friedrichshagen. You'll pass by plenty of beer gardens and ice-cream parlours along the way. Have dinner in the amazing former GDR restaurant-tower called the Müggelturm.
In 2010 the city of Berlin finished its bucolic, riverside stretch of the 671-kilometer-long bike trail between Berlin and Copenhagen. This isn't exactly a weekend trip so better plan ahead. The trail is divided in three stages: the Brandenburger stage (about 150 km), the Mecklenburger stage (250 km) and the Denmark stage (300 km).
The beautiful, rural State of Brandenburg is easily reached by car or public transport. No less than 7000 kilometres of newly built cycle paths (and they're mostly flat) will lead you to the State's key nature attractions. The website 'Brandenburg by Bike' will suggest several ready-make bike tours to choose from.