I drive school bus for the Kearsarge District. It's a good school system, there is a new (his 2nd year) superintendent and he seems great. I have a 6th and 2nd grader in the system. There is a winter activity session after school once a week in Feb./March for about six weeks. There are many choices, one of them is skiing at Mt. Sunapee. My sons are involved in soccer. There is an area soccer league in the fall:
Merrimack Valley Soccer League - New Hampshire Official Website.
The school sponsored sports start in middle school. My older son took karate in Andover, Andover Academy of Self Defense, for three years. The owner/instructor is David Stepp, highly recommended.
Students from "away" should have no trouble blending in. My experience is "new kids" are sought out by existing students looking to find a new friend. We have a lot of transplants here in New London, I suspect Warner isn't much different.
By the way, Market Basket in Warner is the best supermarket for miles around!
Mark