I recently wrote a big long email worth of tips for someone about to move to Chiang Mai. It's stuff that's applicable to most people so I thought I'd repeat a slightly edited version of it here. This is a bit of brain dump but hopefully it makes sense.
There aren't a lot of bad places to live in Chiang Mai. Around the train station is not so great but otherwise its all pretty good. I wouldn't worry too much about where in town to live initially. It's easy to move but more on that later.
Within the old city, which is the area within the moat, it's pretty touristy. The east end of the old city in particular. You'll probably want to avoid that area as its more expensive. Here are two specific accommodation suggestions.
Ban Jed Yod
Ban Jed Yod is a little way away from the center of town although there's plenty of restaurants etc in the neighbourhood. Typically they only rent rooms for 3 months or more. They let us stay for 1 but we had to pay a bit more for the room and we had an existing long term tenant to vouch for us.
Varada is in a more central location although the rooms aren't as big. There's a lot more within walking distance. It's a little bit of a hike but you can walk into the center of town from here.
Finding accommodation is a matter of searching agoda, airbnb plus just wandering around. There's heaps of accommodation that simply isn't listed on the Internet.
Most accommodation is in apartment buildings which are easy to spot due to their height. They almost always have a reception person there during the day, if not 24 hours although the staff there at night frequently don't speak English. You just get yourself to a part of town you want to live in then go into buildings asking to see a room until you find one you like.
You'll usually be paying week by week or month by month so you can move every couple of months if you want. That can be a great way to get to know different parts of town. They may give you the option to sign a lease to get a lower monthly rent. These are still typically quite short, in the region of 3 to maybe 12 months. Some really desirable buildings like Ban Jed Yod may actually require a lease lasting a few months but it's rare. If possible stay somewhere for one week to one month before committing to anything longer.
When you move in they'll want a photocopy of your passport, your signature on some sort of rental agreement specifying what you're paying and what you're getting for it plus the first week or month's rent.
We never had any trouble with getting ripped off with accommodation. I've heard one or two stories about mean hotel operators in the most touristy bits of town but most apartment operators are looking for long term tenants who don't cause trouble and who pay the rent month after month. Listen to your instincts though and if something doesnt feel right just move on.
If you're just not sure where to stay initially consider booking a week at Varada and using that time to look around and decide whether you want to stay there for longer or to move elsewhere. Here are some prices for Varada Place as of late 2012. As you can see paying per month is significantly cheaper than paying per day.
When checking out accommodation it can be handy to bring a smartphone or laptop to check the wifi. More or less 100% of rooms/apartments come with wifi. Unfortunately some buildings are running questionable Internet connection sharing setups. For example, the network is unsecured but when you access anything in a browser you are redirected to a page where you log in and all web traffic is routed through a computer in the apartment building office which filters your traffic.
Typically these are set up to try to prevent people torrenting, which they don't. Beyond the security concerns of having all your web traffic routed through a old Windows XP desktop via a bag of scripts written by some unknown party, often numerous ports are blocked and its just a pain. Of course the net connection goes away every time the Windows desktop locks up.
Ideally you can find somewhere nice to live that doesnt run a set up like this. If you don't know what a port or a torrent is you can probably safely ignore this. Ban Jed Yod is a great place to live but unfortunately they do have this kind of set up. It's the only major down side of living there.
Most restaurants, cafes etc will have free wifi. If your apartment wifi is playing up or you just want a change of scene you can head out, get a banana shake and use the cafe's wifi. It can take some time to find a few nice places but there's no shortage of places to try. There are quite a few around nimmanhaemin rd and in the sois (side streets) to the east of nimmanhaemin. A place called "Chocolate Fact" is particularly nice if you want some coffee and cake while you work although it's not the cheapeast place.
If you happen to need to go to a dentist while you're in town we recommend Grace Dental. They're certainly not the cheapest dentist in Chiang Mai but they have the most modern set up and speak excellent English. They are also likely to be significantly cheaper than a dentist in your home country. They're approximately one third of the price of a dentist in Australia. We tried a few other dentists who were all cheaper but who were not as confidence inspiring.
Finally, if you have time before leaving home it would be great if you could learn to ride a motorcycle. At some point you're going to want to get a scooter to get around town. Chiang Mai really opens up once you have convenient transport. Haggling with songtaews and tuk tuks becomes a drag plus they all tend to vanish in the evening. With a bit of luck you can trawl through thaivisa.com's forums and acquire a cheap scooter/moped for a few hundred dollars that will allow you to get around town.
Personally, I first learned to ride by renting then buying a scooter in Chiang Mai. Quite a few people do this. I only had actual lessons and got a proper motorcycle license when I returned to Australia. While that worked for me and I've never had a serious accident it's not really the smartest way to do it.
Hopefully that helps.