Do I need snow shoes?

Unlikely. However, depending on average snowfall and where you walk, you'll may want waterproof boots that come up to at least your ankle (Gortex or equivalent). This also depends on how much time you plan to spend outside.

Do I need more/better coats than what I have now? (A duster and leather jacket, both fairly thick.)

Depends entirely on how much time you're outside. I tended to walk everywhere in my college days, so I wore an insulated coat. You'll probably want gloves, even if you stay inside your car most of the time. If you're outside a lot, consider a knit hat, especially if there's a lot of wind where you live.

Other things:

• Thick socks are awesome for wearing around your house.
• I like having really cheap fingerless knit gloves for typing inside.
• If you do spend time outside, look into flannel lined jeans or khakis. Thermal underwear works too.
• Heating can cost quite a bit of money; set your thermostat lower in the evenings and load up on blankets.
• At home, drafts are your enemy. If you're in a newer place, I'd assume this shouldn't be as much of a problem. If you're in a place with single pane windows, think about doing putting up window insulation film. In college, this helped with heating costs, but they have the downside of clouding the view.
• If you have your own driveway, you need a snow-shovel. Even if you theoretically have someone paid to shovel the snow.