Using a "Beater" Bike
A "beater" bike is a good way to cope with the winter. You don't have to worry about it getting salty and rusted. You can leave it out in the rain and snow.
You will still have to worry about it getting stolen, though. Even though the bike is cheap, thieves will still take it if they can. Locking and theft prevention should help you keep this from happening.
A beater bike can be anything -- a single speed bike with a coaster brake, a three-speed with internal shifting, or an old bike with derailleur shifting. Anything can work, depending on which of Pittsburgh's hills you have to climb. Your choice will depend on what you have available, or what you run across on Craigslist. But simpler is better. If you're going to be doing little to no maintenance on your beater, you'll want to avoid anything that can get gummed up with the cold -- which is why people gravitate towards single speed or internal gearing. On the other hand, some WD-40 will keep your beater bike running and shifting through the winter. Just hit the points that get wet (brake pivots, pulleys, the cable guide under the bottom bracket if there is one, and the chain, of course). Don't worry too much about cleaning; it's a beater; just keep it lubed.
Snow tires are a good way to make your beater bike more usable in the winter. A good pair of snow tires will likely cost more than your beater is worth, but you can take them off and use them year after year as you replace your bike, if you keep the same rim size.
If you'd like to wear normal shoes or boots to keep your feet warm in winter one way to do this is with Power Grips. They'll help you keep your feet on the pedals and even pull up when you're heading uphill. There are also pedals designed to grip the soles of shoes, like the Grip Kings from Rivendell, or BMX pedals with spikes.
Winterizing Your Bike
The biggest step you can take towards making your normal bike usable in winter is to give it some snow tires. These have heavy tread and, in some cases, studs to give you a grip when you hit an ice patch. There's nothing like the feeling of riding confidently on ice and snow past motorists creeping along the streets, terrified of slipping.
For a good, unbiased source of information on snow tire models, see Peter White.
Snow tires won't work on all bikes. They are wider than normal tires and the heavy tread gives them a bigger diameter. You should check clearance at the brakes, fork, and chainstays before purchasing them, especially since they're quite expensive. You can get the tires at some local bike shops, including Thick and REI.
The expense of the tires is offset by their taking a long time to wear out, since their tread is so heavy and you only use them in the winter. They are annoying enough to ride on (slow, heavy, and noisy if you have studs) that you'll want to take them off as soon as the roads clear. So a pair will last you a few years.
If you can afford only one snow tire, it's best to put it on the front. You'll need it to steer. If the rear tire slips while braking you'll still keep going in the right direction (and if it slips while going uphill you can always walk). In fact, this is a good braking strategy on icy streets, even if you don't have snow tires -- always brake with the rear tire first. If you lose traction, it will drag behind you and still slow you down, while you steer. Braking with the front tire is trickier, since if you lose traction the bike can slip sideways out from under you. You might want to stick a foot out, to give you three points of contact in icy areas.
It can be hard to get a snow tire on a wheel. A lot depends on the bead and rim diameters. If you're having trouble, make sure the tire is seated properly in the rim all the way around opposite the part of the tire that won't get over the rim. Some dish soap on the rim might help with that last little bit. You might need to deflate tire to get it between the brake pads, then reinflate after the wheel is mounted.
Snow Tire Alternatives
The expense of snow tires has led people to search for alternatives. One way to give your tires extra traction is to put zip ties around your rim and tire. This only works if you have disc brakes. The zip ties give you extra traction, much like tire chains on a car.
You can also get actual tire chains, which will work on bikes with rim brakes. There are also other options. You can even make your own snow tires using another tire with heavy tread, some sheet metal screws, and a drill.
Any of these alternatives to snow tires have similar requirements about clearance -- you're going to need plenty of it.
Lights are important in winter because the days are shorter. And cold weather can reduce battery life by half. (Though if you have a helmet-mounted light and an external battery pack you can put it under your jacket to prevent this.) Short battery life is a good reason to consider a dynamo lighting system like the ones sold by Peter White. If you have dynamo lights you'll never find yourself out someplace in the dark, with your batteries suddenly giving out in the cold. You'll just be able to hop on your bike and ride, whenever you want.
Lubrication and Cleaning
The best thing you can do for your bike over the winter is to keep it out of the weather as much as possible (get a saddle cover if yours is leather), and to dry and lubricate it when it gets wet. A bike will hold up very well if you simply keep it oiled. If you have a warm place you can wash it count yourself lucky, and take advantage of it; otherwise, you might stop by a car wash and give it a good power cleaning on warmer days. You can also wash your bike off using a pump sprayer like the ones used to spray garden chemicals.
Be careful not to clean your bike TOO much during the winter. Although it’s good to get the salt off periodically, rinsing it every day can end up getting salt and grime into nooks and crannies that it otherwise wouldn’t enter. Also, if you use the car wash, be very careful -- the powerful spray can easily bypass seals and get into your bearings, where the water will quickly rust everything.
Spots to keep an eye on are the cable guide under the bottom bracket, the cables themselves, the derailleurs, and especially the chain. The cables can be lubricated without being completely removed if you have enough slack to slide the housing down. You can lubricate with a light oil and slide the housing back on. If you have problems shifting your cables are probably sticking; try lubricating before replacing the cable.
The oil most commonly recommended is Tri-Flow. (A cheaper alternative is WD-40, i.e., "Water Displacement formula 40", which seems to be designed for just this purpose. But some folks claim it should never be used on a bike. This is the biggest single point of contention in bike maintenance.) You can wash your chain, etc., with a light spray lube, dry it, and then follow up with a good synthetic "wet" lubricant such as Pedro's, Finish Line Wet Lube, Rock & Roll Gold, or Boeshield T-9. Given the variety, you might just go with what your favorite bike shop recommends, or use a light synthetic motor oil if you're on a budget. The most important thing is to keep things lubricated.
Bike Pittsburgh has prepared a useful guide to winter riding while not spending a lot.
Dress warmly, but try not to overdress, as perspiration make your clothing damp which will drastically reduce its ability to insulate. Dress in layers that can be removed before you overheat. You can also wear wool or synthetic layers next to your skin that will wick away any sweat to mitigate this problem.
For your face, a pair of clear safety glasses is a cheap option that can help you keep your eyes on the road even during rain, snow, and harsh winds. Ski goggles are a heavy duty option that may be helpful during particularly adverse conditions. Balaclavas and facemasks offer near complete facial coverage, but a simple scarf or handkerchief has the advantage of being easily adjustable while riding.
Keeping your fingers together helps them stay warm. If thick gloves or doubling up on gloves is insufficient, consider lobster gloves, mittens, or bar mits.