Hi,Ms.Faye.
I'm Chinese and have been living in Kunming for 24 years.So I think I can give you some good suggestions about living in Kunming.Here is my suggestions:

Language school-I'm not sure that if there was a language school which teaches foreigners to speak Mandarin in Kunming.But I will search one for you in my free time,and I will tell you the answer later.If I coudn't find one I think I can be a teacher if you wish.I used to teach Mandarin when I was studying in Peking University.

Proper time moving to Kunming-It depends on season.In spring(Feb to Apr)the weather is warm and dry.The precipitation is low and the wind is powerful.In summer(May to Aug) the weather is hot(about 28~30 centigrades in the day and 16~19 centigrades in the night)in sunny days and cool(sometimes a little cold)in rainy days.But in summer it rains a lot so the weather is cool in most of times.In Autumn(Sept to Nov)the weather is cool and the precipitation decreases quickly,so I think it's the best time for travelers to come to Kunming.And in winter the weather is warm in sunny days(7~10 centigrades in the night and 18~20 centigrades in the day)but a little cold in rainy days.The temperature will be less than 5 centigrades in the night and 10 centigrades in the day.Luckily,in winter the precipitation is very low,so the cold days will be very limited.

Wish you have a pleasant journey in Kunming!