brakes: Learn to brake using both hands. When you get good, you can automatically modulate the correct brake when a wheel locks up. This is a bit of a simplification, but brake before the turn and coast through it - you'll get better traction and leave the turn at a faster speed.
studs: Studs only help on ice. You'll know if you need them because you'll either be scared or bruised.
chain: Have your LBS check your chain for wear and replace it if necessary (otherwise be prepared to replace worn out gears). Keep the chain clean and rust free. I used WD40 for a commuter bike I rode on heavily salted, gritty, slushy roads with excellent results. There is no need to degrease and relube if you apply it frequently.