Thorsten,
I wish you and your family well regardless of where you move to in the US. Having recently moved back to South Dakota (my home state) from California, I understand how traumatic such a move can be. I, at least, am still in the same country.
I would take
Never Forget's comments with a grain of salt. He lives in Perpetual, Misery, and would like everybody else to live there with him.
South Dakota is really two states in one: eastern South Dakota, and western South Dakota, each of which has their own unique characteristics. Where you decide to live will depend on your personal needs and interests.
I was born in eastern SD, and spent my adolescence there. The eastern half of the state shares much of its heritage, culture, and geography with Minnesota, and is heavily influenced by the economy of Minnesota. Geographically, eastern SD is pretty flat with an economy dominated by farming and services. The weather, like Minnesota, is cold and snowy in winter, and hot and humid in the summer. The autumn's are beautiful - the best time of year. Actually, I found the weather for the months that I spent in Germany to be very much like that of eastern SD.
In the southeast corner of the state, there has been the development of a number of financial and manufacturing businesses. The largest state universities are also in the southeast, with the largest and dominant city in the state - Sioux Falls.
The heritage of eastern SD is dominated by Scandinavian, and Central and Eastern European ancestry. Reflecting that heritage, as well as the influence of Minnesota, the eastern half of the state tends to be fairly liberal from a political perspective. The early settlers in the area tended to almost 'commune' type ethnic living, and the area still has that element of socialist populism to it, along with a mix of traditional religious values.
Frankly, as a kid, I found the eastern part of the state to be pretty boring, and I still do as an adult. So I was delighted to move to Rapid City, in the western half of SD, when I was 13 years old.
The western half of the state tends to be more diverse, both geographcally and culturally, than the eastern half. Western SD is dominated by the Black Hills, a beautiful forested area with vast recreational opportunity. The western SD economy is focused around ranching, services, and tourism, with little in the way of manufacturing activity.
The prairie of western SD is quite a bit more rugged than the plains of the eastern state, and is probably home to more ranching than farming. It is also home to most of the Indian reservation land in the state.
The weather in the western half of the state is much more arid than the eastern half, and particularly in the Black Hills area, temperatures are much milder in both summer and winter.
Culturally, the western half of the state, the Black Hills in particular, are quite a bit more diverse than the eastern state. Originally settled during the Black Hills Gold Rush of the late 1800's, the area attracted a broad mix of cultures in to make their fortune in mining. Today, the dominance of the tourist industry, particularly with Mount Rushmore and the Sturgis Motorcycle Rally, brings in a diverse crowd of people from all over the country and the world. The presence of Ellsworth Air Force Base near Rapid City, also brings in a diverse mix of people.
Of note in Rapid City is the increasing development of the area as a major regional, multi-state medical care center.
Personally, my strong preference for living in South Dakota would be in the Black Hills area. If you are into any sort of recreational activity - fishing, hiking, hunting, camping, mountain-biking, skiing, boating - there are great opportunities to keep you constantly on the go within the huge Black Hills National Forest area.
On the other hand, if you are into the arts, you might prefer the Sioux Falls area.
Cost-wise, I can compare the area favorably to California. Costs for energy - gasoline, electricity, natural gas, oil - are significantly less in SD. Costs for housing are probably one-third the cost in SD than California. Food costs are also less in SD. South Dakota has no income tax, and its sales tax, while higher than some neighboring states, is certainly lower than California and New York. Property taxes, however, tend to be somewhat on the high side. On the downside, the prospects for employment are certainly not as great in SD, and the wages and salaries are probably half what you would get in California.
I hope that gives you a little better idea about what to expect in South Dakota.