What kind of car should I get that works good in the snow for an assignment at Fort Drum?

If you are a trail nut get a 4 wheel drive. If you stick to the streets get an AWD

Was stationed at Drum for 6 1/2 years. Do not get anything with rear wheel drive. Either front, 4, or AWD. Also consider the fact that weight equals traction. The more the vehicle weighs the more traction you'll have. Those cute SUVs are nice but suck in winter driving due to light weight. My recommendation is full size truck or SUV.

The first thing I would say is that if you are unaccustomed to snow, you need to learn how to drive in it before you invest too much in a vehicle. Truly, as an E-1, that is probably a good choice anyway.
Get a vehicle with 4WD. If it is a pickup, get some sandbags to get some weight over the axles.
Invest in decent tires, as they are the only thing between the snowy road and your vehicle.

Once you have procured your vehicle, find an open place like a parking lot to find out what it can and can't do. How long it takes to stop. How to approach a stop. Turning in slippery conditions. Things like that.

If you find that you can't afford what you need, make friends with someone who does and chip in for fuel. Smart money is to have a friend along with you whenever you venture into Watertown or points further afield anyway. Keeps everyone safer.
Good luck, SPC (Join to see)

Any front wheel drive or all wheel drive car will serve you well. I have lived all my life in this region, (Saranac Lake and now Central Vermont), and have driven both. You will not be four wheeling with it, and the highway departments are well equipped and skilled at snow removal. It will not be like Virginia where we have kids and grandkids, and the streets of Fairfax are half-axed plowed at best.

I drove a Honda Accord for years year around with no problems. Currently we have a CRV and I have driven it in 8-10 inches of snow (my driveway) without getting stuck.

You can get snow tires with or without studs. If you go studded they have to be off the vehicle by May 1st in NY. Personally, I drive with good quality all weather tires, and have never had a traction problem.

My caveats would be directed more to the vehicle. Make sure it is properly winterized. What will be OK in more Southern locales, will not cut it in areas where -20F is not unusual. Inspect for rust, especially in an older car. Salt is murder on frames and bodies. Personally I take my car through a car wash with an underbody spray at least once a month in the winter.

If you get a chance visit Saranac Lake during winter carnival. Usually the first full week in February. It is a blast. There are also many world class events in Lake Placid.

Only two things you need for any vehicle in snow are good snow tires and slow down. For a brand new private, I'd say a used subaru in the $5-7k range.

Something with 4 Wheel Drive or all wheel drive.

F150 super crew 4x4

SPC (Join to see) great question first get something you can afford. Next, ask you NCO to offer advice when you get here and lastly do not buy locally get away from the base to purchase any vehicle.