Unfortunately, South Dakota and Minnesota do not have a recipricol state tax agreement.
This means you will have to file taxes in both states, but you will not be double taxed.
Income taxes for Minnesota should have already been withheld, so you will want to file in Minnesota first as a non-resident. Minnesota will be entitled to taxes on all income from Minnesotat sources.
South Dakota will want to tax your world wide income as a resident. However, SD will give you credit for taxes paid to Minnesota, so that you are not doulbe taxed. They will however collect the difference if the tax in SD would be higher than the tax in Mn.