We are coming to Chiang Mai soon!

Your plan might be a little sticky. December though about February or March is the high tourist season. Even Russians vacation here.
Renting a house often requires a 12 month lease. However, there are tons of exceptions. You could start at agoda.com to familiarize yourself with properties and rates. However agoda operates as an agent for rentals for less then 30 days (daily rate). There are at least two agents that often rent to foreigners. One is Chiang Mai Properties. Researching their listings is educational. Another is Perfect Homes. I like Perfect Homes better because their listings are posted on a map, in contrast to simply listing the district. (Very confusing if you don't know the city.)
Ahead of time you can look up the Chiang Mai Expats Club for a variety of information. They have gatherings including a bi monthly breakfast, plus sub groups. You can look up meetup.com to see some other activities. Chiang Mai is a friendly city (with it's own inherit short comings as well). It's not hard to find an open air cafe or coffee or such and enjoy watching people. It's also very easy to turn to the person at the next table and simply ask them where they are from. Almost always you'll start an enjoyable conversation. There are many resources to help you with whatever you want, formal and informal.
Chiang Mai has many health facilities, including recovery or self help. You could probably do a lot of research on line. One complication would be that of a work permit. In order to work or volunteer one needs a work permit. Actually you could contact Assist Thai Visa in Chiang Mai. They are a visa service and the owner is an English lawyer. Try and communicate with the owner Rhys Bonney. Now a days immigration laws are changing quickly.
That should give you a month's worth of work to do. By the way meals can cost from $1 - $150, or more depending upon the occasion. If you get really board you can take a short hop to many places or countries. In January you should check out the Queen's orchid farm, and ask if they are in bloom. Plus the grounds are a sight to see.
There is an ongoing population of around 30,000 expats. Almost all have some working knowledge of English. You can readily get buy with your English for the most part. In the central area everyone has some one around who can help with English, or actually help with speaking English.

You'll probably enjoy the many discoveries that Chiang Mai has to offer.

Enjoy

Thanks for all the good information, Bill.  I'll be sure to check out the Queen's orchid farm. I had stumbled on Agoda in my searching, and I'll get back into it now that travel dates are set. It looked like there was lots of availability during our visit, so I hope that's right. We will probably stay in two or three different places to get a feel for a few different areas. What do you know about Chiang Mai Buddy? I am tempted to use some of their services, such as airport pick up and a tour of the area, unless I hear of something better. I don't want the house for this visit, but want to know what areas to look in in the event we really do decide to move there.  Thanks especially for the names of people and places you think well enough of to recommend.

To answer the question simply, I am not personally familiar with Chiang Mai Buddy. The airport does have airport taxi's and other means of getting to your lodging, and the airport is in the city which means it's close to many things. Since you will be a newbie Buddy might be a means of familiarizing yourself with the city. Plus there are many tourist offices that can help you, some public.
My first trip here, four years ago, I boarded my dog while I was here. Since you are making an exploratory trip it might be helpful if you do the same. Bringing pets might create complications and limit some of your options. I didn't want to board my dog but I thought that it was expedient to do so with an exploratory trip. Many landlords do not want pets in their property, in part because of the other guests. Please, do not be overly friendly with any stray dog in Thailand. You just never know about the dog and there's still a fair about of rabies in the country. Just a little advice. 
One concept to think of is that the further you get out of the city the greater the tendency for your language and survival skills will be challenged. I live in a village that is about 15/20 minutes from the center of town. Within the area there are shops that provide immediate needs, such as 7-11 or local shops. But I have my own transportation.
There is something called CMCC or Chiang Mai Community Church. I think you can find them on line. They are a Christian organization that provides transitioning help for missionaries and expats. They conduct classes and have printed material, and is centrally located.
By the way - Niemannheiman (SP) road is the upscale part of the city where many "Farrongs" (foreigner) reside and hang out. I'm not sure if that's a recommendation or a warning. I think many foreigners start their exploration in the proximity of the inner city. Part of that is that you can walk to most places and the greatest concentration of English speakers is there. Then most start exploring outer areas.
Be for warned - driving is very dangerous for a foreigner. There is a flow and spontaneity with Thai drivers. Add tourist driving motor bikes and it adds a little more excitement to the mix. For short trips you can use Sung Tows (phonetically spelled) or red trucks. They are Toyota trucks with benches and they act something like a bus. You tell the driver where you want to go and he/she will tell you if they will go there. Plus, in an effort to clarify, you establish you price before getting on the truck. Typically it's 20/40 baht per person.
Oh socially there's another little item. Thai's are very agreeable. When asking for information a Thai will typically give an affirmation. In part they do not want you to know that they don't know the answer. Not keeping this in the back of your mind can create some very confusing situations. Every one who comes to Thailand learns that there is - The Thai Way.
How's that for a little more information?

Enjoy,

Trudette, how was Chiang Mai?