OLD CITY - Cross the street (the moat) and you are inside the old city. Stroll around and get lost, you will stumble upon coffeeshops, restaurants, temples....
CHANG MOI - take a turn left-left when you leave the house and you will be strolling around Chang Moi, upcoming area with hip coffeeshops, here and there fashion. End up at China Town, especially in the late afternoon nice area because of lot of shadow, lots of shops.
NIMMAN - the hip modern part of Chiang Mai, west of the moat. Especially young Thai and students love to hang out there, as well as Expats. Fancy international restaurants also to be found here, Italian, Japanese/Sushi and so on.