It is easy to spot Chiang Mai's city on a map as it is clearly delinetated by the ancient city walls that once surrounded the fortified old town. Chiang Mai City is considered to be everything within the walls and the shops, restaurants and hotels than run alongside the canals that ring it. 

There is a rather busy and confusing ring road that surrounds the city, but once you are inside the walls it is far calmer, with plenty of tiny lanes and alleys to explore. This is also where you will find many of the temples including Wat Phra Singh,Wat Chiang Man and Wat Chedi Luang. The area is just about walkable if you are fit, but there are plenty of tuk tuks and red songtaews if you get tired.