We are going to put in a 100 amp sub panel from a pole barn that has a 200 amp panel. This 100 amp panel will be 125 feet way in another pole barn. My question is what kind of wire do we need if it is going to be buried. This is in Northern Minnesota. Thank you

I would Use 3 # 2's. Could probably go down 1 size with neutral. A #6 for Ground, this would be for Copper.
I would Use PVC. 2" will not hurt, may help in the future. Trench is the Money.
Throw in 3/4 PVC conduits also for TV, SAT, Phone, Alarm, Innercomm, 3 Way Switching for lights?