Welcome!
We recently moved BACK to MI from KY after 6 years. I forgot how cold it can get!!! We are always on a budget so hopefully I can help here. It'll be long so bare with me. I may repeat myself too...sorry if I do!
Make sure you ALL have warm boots and socks. The kids will need snow pants if you plan to go sledding (you too if you don't want to get wet)...ask new neighbors for the best hills. You'll need heavy coats too along with the obvious hats, gloves, and scarves. I like the waterproof gloves to play in the snow. Other's will get wet and hands will get cold. Sometimes I'll layer the cloth gloves inside the waterproof for extra warmth. Waterproof coats are a must if playing in the snow.
It can get bitter here in the winter and my youngest daughter gets chapped cheeks. If she's out in the cold for a minute with bare cheeks she gets really red and chapped. I like to use a face stick. Aveeno is the only one I have found at grocery stores. It just keeps the skin protected when the kids are outside and helps to heal skin if it's already chapped.
CHAPSTICK! Can't stress that one enough. Apply often. Pretty much any brand will do. I'll even use vaselene if I can't find lip balm.
The air will be dry so you will need plenty of lotion. Even the kids. Again...anything will do as long as you are applying it daily...sometimes more than once a day. Drink plenty of water as well to keep skin hydrated. Otherwise you run the risk of itchy, cracked, scaly, ashy skin. Like a gator!
A humidifier can be a great help to keep air moist in the home...it will feel much colder if the air is dry...I know because we went without one last year...big mistake!!! We were renting an apartment and the windows were drafty. Spent all winter in a blanket!
Static guard will be your best friend. Because of the dry air you will collect all kinds of static resulting in electric shock every time you touch metal or another person. Seriously...I have whitnessed an actual BOLT come from my finger. Does not feel good. That reminds me...when pumping gas in this dry air be sure to "shock" yourself by touching your car before you touch the pump. Try your best to keep one hand on the car at all times...this will help you avoid charging the gas pump and causing a fire. It's happened...look it up. Also...reguarding the static guard, spray it on everything...hats, gloves, coats, car seats, blankets, anything that collects static. I like to spray a bit on my hairbrush to avoid flyaways. Your skirt, pants, shirts etc. will stick to you and other things. If you are wearing a sweater over a T-shirt be careful when taking off that sweatshirt in front of people. I once had an unfortunate experience where my t-shirt came up along with my sweatshirt giving everyone a show. Luckily I was among friends and it was just plain funny. But you can see how this could have gone in a whole different direction!
I don't know about shopping in Ann Arbor however if there is a Walmart shop there. Usually they have the best prices and if you don't plan on needing this stuff for too long you won't need that expensive tripple stitched down jacket at the mall. I purchased a great winter jacket, boots, and snowpants at Walmart last year for my oldest daughter and she loves them. All of it is still in great shape and will be used this year as well. We also purchased two sleds for $4 a piece too. Not bad! Get them early though because once it snows...they are hard to find. Target isn't too bad either but can be a tad more expensive. You can also look for mom-to-mom sales or resale shops. I've never been (i'm too lazy) but I know plenty of people who have found all kinds of great items for next to nothing.
I don't care for the look of turtle necks so we don't wear them around our house. I just make sure the kids are well bundled with their coat, hat, and scarf to cover their neck. Long sleeves are usually a must for kids because they get cold quicker. For me...I get hot fast so I like to wear layers or a 3/4 length sleeve shirt. If you find your kids get hot fast too...again layers are always a good idea. I'll usually do long sleeve under short sleeve but that's just because I like the way it looks. If my kids get hot they can take off the long sleeve or the short one. Whichever one they want.
Jeans and sweats are perfect. You may want to get one or two pairs of long underware for everyone. Last year we had EXTREMELY cold temperetures. I just would throw on a pair of tight running pants under my jeans (because that's what I had). My hubby would wear his pj pants under his pants, and my kids would wear their pj pants too. They have a few pairs that taper at the end so they were perfect.
That's all I can think of for now. I'm sure there is plenty I missed! If I remember I'll update for you.
Good luck this winter! Hope you have fun!