Here are several longer rides that I have done/liked so far
-Anywhere along the Tama river, though I probably would not attempt it on a weekend due to pedestrian traffic.
-There is a bicycle path that starts at the end of Inokashira Dori (just north of Musashi-Sakai station) that leads to a large water reservoir with a nice looped bicycle path around it north of Tachikawa
-Take Route 1/Tokaido south so as far as you like (Chigasaki/Odawara/Hakone)
As far as riding around the city I do not really find other motorists too much of hassle except for on those rare occasions of crazy taxi cab drivers/people on scooters riding too close to the shoulder/people on mama-chari's riding against traffic.