First of all, remember it will take your body a couple weeks to acclimatize to the cold and you will need to hydrate and fuel more. I've spent several winters in Northern New Hampshire training for Boston and never once ran on treadmill, I took my cue from Nordic skiers for layering before eventually just switching to that sport for the winter. I also lead winter camping trips for work, which have included tarp camping with one trip being a week of lows at -30F and highs of 10F. Remember that you should feel chilly at the start, and that sweat=cold. The following tips are based on my experience but will probably work for you too:
Head layer: A thin headband will keep your ears warm and allow you to regulate heat/not sweat, a fleecy headband is great when it's colder/windier. A thin hat when it's super cold. A thin buff for your neck is really helpful too, the thick fleecy ones will be too hot. A few dabs of vasoline or dermatone on your nose/cheeks when it's windy will help keep frostbite away
Shirts: Thin baselayer, wool works well, anything too thick or fleecy will hold sweat. Wool or synthetic shirt over baselayer, definitely no cotton. Lightweight quarterzip as your outer layer, this should be breathable, having a zip will allow you to dump heat when you are too hot so you can avoid sweating too much. A layer with a wind blocking front but breathable back is key for windy days.
Bottoms: : Some companies like Craft and Smartwool make briefs with windblocking over the crotch, these are well worth the investment. Running pants/tights based on preference. Base tights for under your pants/tights on colder days. Consider some xc ski pants/tights/base tights with wind blocking on thighs for those super windy days.
Hands: Thin running gloves, thicker xc ski gloves, thick mittens, it all depends on temp. There are some mornings when I will wear thin gloves under puffy mittens that I can pack away really small when my hands get hot.
Feet: Wool socks.
Major takeaways are no cotton ever and don't wear things that will keep moisture in. Traction is a personal choice, you can go blue collar, yack tracks, or studded shoes. Any good running/xc ski store up there should point you in the right direction for gear. Get out there and enjoy winter, it's a beautiful time of year.