From the pic you showed it looks like the sills are barely above ground. It looks like you have a good slope all around. at least on the down hill side. You are in a cold climate and good grading is important. A french drain all around where the drip line from the roof. I don't know what temperatures you will be dealing with. Once heat is in the barn, i think the slope to the lower roof will shed snow and the ice buildup will not create a dam at the eaves. i don't think your should install gutters. You have good grade and a good french drain to grade should be adequate. Better make the surface of the stone a little lower than grade to catch water from snowmelt and rain. From your photo, it looks like you may have vertical siding. Whatever siding you have, see if it is possible to tuck flashing under the siding and down at least to ground level along the foundation. It you can afford to regrade around the foundation, get the gradeatleast afoot down from the sills. That might not be possible if the footings are not deep enough.
When i was in construction, it is funny, i almost never was called to fix water problem when a house was close to a swamp. All the problems were on high ground like your barn is. The responder who suggested don't try to get rid of the water after it gets into the building, just figure out how to get it away from the building before it gets in. you had good grade on the one side of the barn. I can't tell what the other side is, but you do have change to drain the water well awayfrom the building. I don't recommend the gutters in your climate asi mentioned above. I have had to shovel many roofs in heavy snow year. All had gutters. And, they were way down here in Connecticut. I think you just might have a little more snow in your area. Maybe not so much ice involvement with winter storms. I don't know about that.
Stay away from dry wells to catch all the water from the roof unless you have excellent drainage. Heavy rain after the gound is already saturated from week of rain, and all the wells can be full and then you have more problems.
Tinker