This post is also under the page nuts'n bolts, cups'n cones-
The most integral part to a bike as a whole, is the frame. The rest is a conglomeration of pieces and parts (some of which work better than others). In summation, they all work together to create a simple machine for transportation. You put in the energy and the wheels will take you wherever you want to go!
Frames are made of different materials, the main 3 being steel, aluminum alloy, and what's known as carbon (which is really a composite of woven sheets of carbon fiber and apoxy of various strengths, stiffness, and weight). Within these 3 categories there are multiple variances that determine the way your frame feels (stiffness, weight, steering) when you ride it. Some are more comfortable than others- the winner generally being steel for its durability and soft, reliable ride. Aluminum became really popular in the mid-80's to early 90's as a frame material because it is slightly lighter than steel and a lot stiffer, which makes the bike ride feel faster, esp. in handling. Mid-90's, carbon also started making it into higher end bicycles and is now kind of the standard for anything 'racy'. Alright, where does that leave your frame?
If you've never serviced your frame with an overhaul- now's the time. You will be able to determine several things. If you're feeling the wobbles when you ride, you probably have a loose hub on your wheel, a loose headset, or even a loose bottom bracket. The headset and bottom bracket are not part of your frame. They are parts attached to your frame either by threading on the frame or the fork. Take out the BB, seatpost, and the headset and check to see if the threading is damaged in any way (if they have siezed, they will be impossible to remove w/o significant effort and possibly ruining the frame). If the threading is damaged, you can chase it (although, the fork is better replaced with a new one. Once those threads are gone, it will change the diameter of of the stearer-tube. It is easiest and safest to replace the fork with one that has the same rake and diameter (either 1" or 1 1/8"). Take this opportunity to replace your BB and headset (each about $20 for a standard, cheapo replacement) so you have new ball-bearings and races. This will make your ride smoother and safer!
While you have your frame open take the opportunity, again, to observe a few things that effect the integrity of your frame. Look for rust or corrosion at the weep-holes or inside the tubing. If the inside of your frame is starting to rust, it could very well be deeper that you think and your frame probably has a limited life expectancy. If your frame is not steel, look for signs of cracking in the paint. If your bike has ever been crashed, pay close attention to damage that may not be immediately visible. Also check the exterior of your frame for dings, divits, or large patches missing paint (esp. if they've started to rust). Surface rust can be polished off with steel wool and then given a base-coat of matching fingernail polish or even clear polish. This will keep the elements from continuing to eat your metal!
Sometimes, even when the damage isn't that bad, you should assess your bike from the standpoint of function and and cost to fix it and make it safe to ride. If your drive-train is not functioning safely and you need to replace it, this would be the time. Usually it's easiest and cheapest to convert to a single-speed. If your brakes aren't the best, take them off, clean them, check the springs. If you can find good replacements, you're lucky. Older frames are drilled for a different type of brake than is standard now, so if you want to get new brakes, you may be looking at compatibility issues.
If repairing the bicycle is going to cost MORE than the bike is worth, or what you can afford- sometimes it is best to let the poor beast go and either get a new frame in better condition, or start saving for a new bike altogether. It is not safe to continue riding a bicycle that has been compromised unless you are willing to put in the time, effort, and money to continuously care for it and wait for the day when it's no longer tenable.
One more option comes to mind, since you all live in a wintery place. Having a winter bike is a great idea- use your oldest bike or the one in question and don't worry about the fact that after riding it all winter it will probably be finished.