It seems that over the past few year, my post on winter running has been viewed quite a few times. Starting to run in the winter in Minnesota is no small endeavor. My previous post on winter running can be viewed here, but feel free to read more specific advice below.
Although running is a relatively cheap winter sport compared to, say, skiing, winter running still requires an investment. The essentials for winter running in Minnesota are clothing and proper footwear.
Clothing can be pretty simple. A nice pair of running pants works well for temperatures between 10 and 40 degrees. Down in the single digits and colder, however, adding another layer to running pants will probably be necessary--tights are a good choice, but be sure to steer clear of anything cotton. As soon as cotton gets wet it loses just about all of its insulating ability.
On the top a good long-sleeve tech shirt and a light running jacket work well. The colder it is, the more layers you can put on. When it gets really cold, I recommend the following layers, from inside to out: short sleeve tech shirt, long sleeve tech shirt, windbreaker, medium-heavy fleece. Having the windbreaker on in between layers does a really nice job of keeping the core warm.
A good hat and pair of gloves are also essential. Again, cotton is a poor choice--there are lots of good products out there made of polypropylene or other synthetic materials that wick away moisture and keep your head and hands warm.
Winter footwear is perhaps the trickiest. A pair of insulated trail running shoes can do the trick on some surfaces, but for more snow and ice, you're going to need some sort of traction product. Yaktrax work fairly well, as do these cheaper, do-it-yourself screw shoes. I find screw shoes work a little better on ice and Yaktrax a little better on snow.
The other thing to be aware of in winter running in the northern climes is decreased daylight hours. Running outdoors in the early mornings or in the evening usually means running in the dark. If you live in a well-let area, you may be able to get away with just wearing reflective gear, but I would still recommend some sort of headlamp. I actually have a cheap Eddy Bauer headlamp from Target (though I don't believe they sell it anymore), and it works just fine. There are plenty of options out there, but be sure to go with something lightweight and water resistant.
There you have it--everything you need to know about winter running up north.