Chiang Mai believe it or not has a better selection of guest houses than Bangkok. You will find them to be friendlier, laidback, quieter and more often than not have a better atmosphere. Choose from shared dormatory type rooms from 100 baht per night up to 400 baht per night rooms with own bathroom, etc. Many of the guest houses will try to pressure you into taking a trek into the mountains with them, they do this to off set the low rental of the rooms. If trekking is not your scene, just say you have a bad back! Most of the cheaper accomodation in Chiang Mai city is conviently located around the eastern site of the moat in an area called Tha Pae Gate, which has a good selection of places to eat and market stalls.
Starting at around 500 baht buys you something that little bit better. Either good quality guest houses or small hotels can be found at this rate offering hot water, tv, air-con and an overall better standard. These type of places often have a coffee shop where you can chill out and talk with fellow travellers.
Over the past couple of years high end accomodation has been popping up all over Chiang Mai. These range from unique boutique hotels right upto massive 5* resorts. These hotels offer every luxury, so if you can pamper yourself.