Running in the cold is perfectly fine, and I've run all the days of this season, including when it was down to 15°F so far, with not a ton of extra equipment. I don't know how recommended it is if you are going to be travelling from the southern weather winter to a northern weather winter suddenly might be a shock to the system.
Protect your extremities more than you think you should. In 30°F weather, for example, I am running in shorts and a long-sleeve shirt, but I also wear a hat and gloves. If it's windy, I also wear a windbreaker over my top.
If it's really cold, think in Layers. A Compression top is great to wear under your normal running tee, it will block a lot of the wind and also provide some insulation while wicking away the sweat and keeping you dry. The same goes for the bottoms - get a set of running tights, which will normally work fine down to about 0°-10°F, and then add some shorts on top of them to block wind, and if it's really cold and you want to protect your man-parts, get some thermal boxer shorts too, that block the wind even more. It's pretty easy to take layers off if you get too hot when you're running, but frostbite is a serious concern when it gets below 20°F.
Don't run in only cotton or another non-wicking material when it's cold, it is dangerous. Your sweat will freeze and you can get Hypothermia. Wool is better, and even then I use polyester wicking fabrics as the base layer.
Running on ice is tricky, go slower than you expect, and only do it in the daylight. I have never trusted any light to keep a path lit enough for making sure that you have sure footing when there is a lot of ice. You can get some Yaktrax to go over your normal running shoes, or if you have an old pair of shoes that you are willing to sacrifice, there is nothing better than putting some Sheet Metal Screws directly into the shoe.
Speaking of light, it gets very dark very early in the northern states in the winter. Here in Minnesota, sunset is before 5pm already (mid-november) and I expect it to be almost 4pm later this year. If there isn't ice outside, you will need some lights and/or reflective strips so that you are visible to cars and other runners while you run.
I don't expect to fall back on the treadmill option unless it's seriously snowy outside, or my streets and trails have more than a foot of snow on them and haven't been plowed. I don't usually worry about rain in the summer, and just run outside unless it is really coming down and I can't see.
Wind can be a problem, especially if you are running around a lake where the wind can pick up considerably over the sheet of ice that will form in December. Outer layers that are of a wind-breaking material like Goretex or just a standard wind-blocking nylon can make all the difference.