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

Most cars work just fine once you put snow tires on them. They have little spikes that bite into the snow and give you a good traction. Just stay away from anything with rear wheel drive

As a private you are not going to be making a bunch of money.
I encourage you to go to college. I imagine you'll be stationed at drum for 2 years. You can easily get an associates in that time. You can get all the points for school, that you'll need for E-5 promotion, in that time. There's still going to be plenty of partying on base. Make friends with other guys and gals in the barracks. Carpool with one of them.
Also, not too much further north there is a place called Alexandria Bay. It's beautiful. Not too far south there is syracuse. It's ok. Basically you are close enough to catch an uber to some pretty good spots without spending a load on a car. With a car you're going to have the payment, insurance, gas, and maintenance. E-1-E-4 money is enough to get a car, but not enough to have fun with one. Do your best to not max out credit cards and get yourself into a heap of debt. That will affect you negatively throughout your military career. Budgeting is very important. Learn it now before you become one of the "statistics" with $100,000 of debt and little to show for it.
Also, there are hundreds of state parks in NY. An excellent location if you enjoy hiking or fishing.

DO NOT GET A MUSTANG WITH 24% INTEREST! But seriously check with your first line leader. They should be able to guide you.