You can go west to Nikotama on Kamazawa or Meguro Doori. Those streets aren't very nice, but I've seen worse. The neighborhood of Gakugei-Daigaku (Highshool area) is OK. You wanna stay away from Kan-Nana or Kan-Pachi, but they let you move about without getting lost. I prefer Yamanote and Meiji Doori because I am located farther east to you. (Those kinda stink too)
Be prepared to MASH pedestrian paths. Cops won't stop you if you ride respectably safe, i.e. not hitting pedestrians, not going full-speed. If that don't work, you can take up the road like any vehicle. Drivers are nice and they almost always "reluctantly" give you space. I usually get out of the way for em as soon as I am out of the jam. I feel bad for those people who's just trying to get through the day. (That also means, I always yield to messengers. But then you'll never seem em outside of the financial districts like Ootemachi. So that's not much of trouble anyway.)
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.
Dude, Inokashira-doori is a pretty cool idea. Thanks, MXS. I had a need for a route either from or to Tachikawa area. ... But that street got two parks and go down north to meet the Tama-river. I am gonna do that when it dries up.