Clean and repack your bearings every fall. Most bicycle greases are fine for low temperatures, but if you are biking in sub-zero temperatures you may want to look at a product like Lubripate Mag-1 which is rated for use in temperatures down to -60 degrees Fahrenheit.
If you are biking in wet conditions as well as cold, you may want to clean and repack the bearings in the middle of the cold season or more frequently.
This would include hubs, headsets bottom brackets and pedals if any of them have serviceable rather than sealed bearings.
I have heard that freehubs can be winterized too, but I have never ridden in conditions cold enough to freeze a free hub.
If you have an oil-damped suspension you will likely want to talk to the mechanics at your local bike shop about winterizing the shocks. Low temps will impact the viscosity of the oil and could damage the shocks. If your shocks use elastomer suspensions they will likely freeze and stop working as you approach 0 F, but the bike should still be ridable.
If you use dry or wax lube, you will want to switch to a wet lube for the winter. This will protect your drive train from salt and water a little better, but will require you to spend more time cleaning and re-lubricating the chain.
For snowy conditions, wider tires with widely spaced knobs will give you the best traction. On a MTB you are probably fine with your normal knobby tires. On a road bike you will likely want to switch to cyclocross tires if you have sufficient clearance between the tire and the frame.
If you are frequently biking on ice, you will want to look into studded bicycle tires, which I know are available for mountain bikes, but I haven't seen in my area for road bikes.
These are a couple of things I do to prepare for winter ridding.
Cover tool handles with foam or electrical tape to insulate them. Few things are worse than taking your gloves off to manipulate a tool and having it freeze to your fingers.
Pack extra tubes and a CO2 inflator so that in the case of a flat tire you can change it quicker.
Anything plastic will get brittle and break very easily. So when looking for parts, water bottle cages and toe clips (if you use them) look for metal ones.
If you can, leave your bike in a cold area. If you take a bike out of your warm house into a cold and snowy area, any snow kicked up will melt first on the warm frame and components, then freeze to the bike causing ice buildup. This is particularly annoying on the drivetrain. By the same token, when you bring the bike inside after riding, take care to dry it completely, or any remaining moisture will re-freeze as soon as you take it back out on a ride.