My husband and I have made several January/February trips to Thailand over the past few years. I think that your plan is a good one...a week in Bangkok and there is plenty to do in Chiang Mai for one month. Chiang Rai is a smaller city but I think that a week there would be nice. For one trip we spend a week in Hong Kong and flew from Chiang Mai on Air Asia (a low cost, no frills airline.)
We also settle in and travel at a slower pace.
Transportation:
We have gone from Chiang Mai to Chaing Rai which is a nice trip - we have never rented a car but usually take a VIP Green bus from Chiang Mai to Chaing Rai (though depending on the roads, it can sometimes be a rough trip). We use a Grab Car (like Uber) for getting around Chiang Mai and also for shorter day trips - I believe that you could also hire a Grab car from Chiang Mai to Chiang Rai.) We have not taken the train from Bangkok to Chiang Mai but either fly directly to Chiang Mai (via Seoul) from the US or take a short flight from Bangkok.
Where We Stay in Chiang Mai:
We have stayed at least 4 times at Sakorn Residence Hotel. For shorter trips, we stay in a room, but for one month or longer, we stay in an apartment. It's been great for us. We like the neighborhood near a Ket Wat Temple and it's a short walk to the bridge across the river that goes to a major shopping area. It is also convenient to the Rimping Supermarket where we buy groceries (things we might buy at home that aren't available in smaller local shops) and large bottles of water as you can't drink the local water. When you spend a certain amount (around 30 USD) they offer free delivery.
Things to Do:
There are two things to consider when scheduling...Chinese New Year is a fun event with lots of activities - it's February 7 in 2027. There is also a wonderful Flower Festival Parade that we enjoy - check the date but it's usually the first weekend in February and may be around the same time as Chinese New Year in 2027. I've taken yoga classes and also watercolor paint classes.
Considerations:
The air quality is not good at times so if you're sensitive, check into the the current situation for the early winter - it depends on the area but that's something to research.
