I'd say that staying within the walls of the old city is a good start for any new comers visiting Chiang Mai, especially if you're a good walker, as it's an interesting part of town full of history & creativity, with a lot of the action being located down the small sois & there's just so much more to see when you're on foot.
But once you've got used to the place & if you like to have your own transport at hand, I'd recommend getting yourself either somewhere along the river or on Doi Suthep mountain for a better experience.
Essentially, Chiang Mai is far more about the scenic aspects of the region than it is the city centre, which can get pretty old after a fairly short amount of time & in addition to that much of Chiang Mai is now being rebuilt around the three outer ring roads & both businesses & potential residents are being given cash incentives to relocate there, so that's now where you'll find many of the larger restaurants, nightclubs & bars.
Do you think I could be writing travel guides?