Welcome to Minot man. The weekends are pretty empty for me, and I've been here for about a year and a half. I'd recommend you pick up a hobby or two really quick. Since I've gotten up here I've gotten into photography, astrophotography, flying RC things, and I even bought a digital piano for my dorm room I just bought a nice road bicycle last week, and I'm super excited to get in with the riding community here. Most of my friends just spend their money on computers/consoles, games, guns, cars, and alcohol, but those sorts of thing doesn't interest me. It isn't easy branching out, but I've found some interesting things in the area!
The Maysa Arena has public skating. I had never ice skated before getting up here and I didn't give it much interest, but it's really fun and I highly recommend you give it a shot even if you know you're horrible at it. A lot of young locals go here so it's a good way to make new friends in the local area. Here's their schedule.
Strawberry Lake campground! Settled in the Turtle "Mountains" about an hour and a half north of town, and it's a good place to get away for the weekend. Here's a picture I took there.
There's a thing called the Denbigh Experimental Forest a little ways to the east here. Experimental forests were the result of a grand plan starting in 1908 to basically plant a forest in a line from North Dakota to Northern Texas to help eliminate wind erosion and provide jobs for people during the Great Depression. This never happened, and it turned into a way to see what kinds of trees would grow here in ND. I haven't been there personally, but some friends have hiked around there and have had good things to say about it.
In the winter, there are two skiing opportunities in the semi-local area. Bottineau Winter Park to the north, and Huff Hills to the south. If you're from anywhere with mountains like me, these places aren't going to be that great. I will say that both locations were better than I expected, but that isn't saying much! Still a great weekend activity to learn how to ski or snowboard, or to teach a friend.
For summer activities, I know a lot of people who go out boating a lot. We have a lot of huge lakes in the area, and they attract a lot of locals too so it's a good way to meet new people. Bonfires are popular up here too.
Definitely think about making trips to either the Roosevelt National Park or the Badlands. Definitely the most interesting locations in North Dakota geographically.
I know I have a lot more activity ideas, but I'll add them when I think of them. What brings you to Minot? Military, oil, the economy, or something else? A lot of people give Minot a bad rap but it's really not that bad of a location. If you have any questions just ask.