On most of the 12 kilometer journey it runs along its own tracks; but it also runs along Meiji Dori between Asukayama Station and Oji Eki-mae Station. The entire ride takes about 50 minutes. It is a slow, and leisurely ride through some interesting sections of Tokyo's 'Shita-machi' (the traditional 'downtown').
Some interesting stations for explorations include:*Waseda - home of Waseda University, one of the top private universities in Japan. The campus is open to the public and is covered with many trees.
*Zoshigaya - Zoshigaya Cemetery and Gokokuji Temple
*Koshinzuka - Sugamo, the 'Harajuku' of the elderly
*Asukayama - Asukayama Park and it's museums
*Arakawa-yuenchimae - Children's amusement park
*Minowabashi - very 'downtown' atmosphere with once popular shotengai covered shopping arcade
A ride along the Arakawa Line costs a flat 160 yen, no matter how far you go. I recommend buying the 'Ichinichi Josya Ken' for 400 yen. With this ticket, you can hop on and off the train as much as you like for one day.
To see a map of the streetcar (with connecting train lines), use this link:
Exploring along the Arakawa Line is a great, inexpensive way to spend the day!