I just came back from Chiang Mai 2 days ago!
Best time to visit is Nov-Jan, where weather is cool and dry (no rain). In Northern Thailand, Nov is the sunflower bloom period, while Jan/Feb is the Sakura bloom period. You won't find them inside Chiangmai, but when you travel out to places like Mae Ho and Mae Chaem, you can visit the sunflower field. For Sakura, do visit Doi Mae Salong, and Doi Ang Khang.
1) Temples in Chiangmai worth visiting - Wat Chedi Luang, Wat Phrah Singh, and Wat Doi Suthep.
2) Shopping in Chiangmai - Sunday walking street (more than 2km of crazy shopping that will take you at least 3hrs! Also do visit the more commercialised Night Bazaar on other days.
3) Short day trips outside Chiangmai - Rent a car/bike and do the Sameong Loop, can stop by at Mae Sa Valley, Sankampeng, and Bo Sang Umbrella factories. Can also do Doi Inthanon (highest mountain in Thailand), or onto ObLuang Canyon.
4) Overnight trips out of Chiangmai - Visit Golden Triangle (take a boat ride on the Mekong), Mae Sai (border town with cheap stuffs), Doi Tung, Doi Mae Salong, Doi Ang Khang, Doi Phukha, Pai (beautiful town and surroundings), as well as Mae Hong Son (for the long neck Karens).