Keep an eye on the rim brakes and make sure they lie flat and line up with the wheel rim regularly. Avoid getting lubricants on the rim during maintenance and if you do, clean the rim with a degreaser (windex will do) since you grease up that rim, you ain't stoppin' for nobody.
Regularly re-lubricate the chain. As mentioned above 3-in-1 is good for fair weather riding, you can also use some of the simple lubricants like "Liquid Wrench" -- just avoid anything that says "penetrating" or that is a cleaner -- like WD-40 or PB-50 -- as that can seep into the areas that are greased and ... well, wash away that grease leaving you with no lubrication. Do NOT apply excessive amounts of lubrication to the spindles (the center part of a wheel that is bolted to the frame) as that can also seep in and wash away the grease...
... and no grease inside the bearings can quickly lead to mechanical failure. One little drop of oil is usually all you need there.
If you ride wet or get stuck in the rain, pat the bike dry when you get home, allow to air dry for at least 8 hours, then apply a drop of oil to the spindle mounts and re-lube the chain.
If you check your tire pressures on hot days (>75F/24C) when the air inside the tire has expanded and the rubber is softer, you want to measure/inflate to five pounds over the rating listed on the side of the tire! Otherwise during the first cold snap you ride you'll be under-inflated! I suggest testing tire pressure before EVERY ride. A tire that 'looks ok' or 'feels ok' can be as much as 15 pounds under-inflated -- resulting in the bike being more prone to fishtailing if you hit a patch of loose earth and significantly more prone to
If you get a flat, do NOT ride it on the rim; that will just bend the rim being something ELSE you have to spend to replace; and a LOT of shops won't lace a new rim onto hubs anymore or even have a truing stand -- which means you're looking at buying a whole new wheel. (wheel = hub + spokes + rim + any sprockets). Even more dangerous if you're a fat ****... like me, you risk splitting the weld on the rim; a catastrophic failure that ... well... isn't pretty.
If you have a multi-speed de-railer type bike (which most multi-speed bikes are de-railer types; basically jumping the chain from gear to gear in mid-air by applying shearing force to it -- such a SOUND engineering design) check that the tension arm (the part that dips down below the hub to take up slack in the chain) is properly moving. This is one item you can go high-hog on with the oils. Because it hangs down farther it's the location where the most gunk -- road dust, water and mud from puddles -- tends to gather so a proper cleaning and lubrication of that is something you should really do every... eh, 50 miles/3 months or so of fair weather riding (whichever comes first).
To that end, I can't emphasize enough how handy a odometer is. Knowing how far you've gone can help you set up a maintenance routine if you ride frequently.
Also, NEVER ride a bike that's just "sat there" for several months without checking that the chain moves freely, the various cables for things like brakes move freely, and that there's no 'grinding'. Lubricants can seep out and dust can collect on it; a good soap and water cleaning with a sponge, brisk rinse with a high pressure hose, pat dry with a towel, air dry (at least 2 hours) and then re-lubrication of the chain and single drops of oil at all the spindle ends is often a good idea. It's the most basic of maintenance and often the most neglected.
Case in point I did that today with mine; even whipped out the white-wall cleaner to get them back to shocking white -- pretty cool given they'd yellowed from all the road salt over the winter.
Since cold temperatures and snowstorms don't really deter me a whole lot; I can dress warm windproof leathers with ski-mask and I've got a set of homebrew spiked tires I can switch to. (which are my old road set with penny nails melted in with a second busted tube preventing them from damaging the normal tube).
You'll notice I talk a LOT about lubing the chain -- it's the most common point of failure and often the most poorly maintained item. There's really no such thing as overlubricating (at least until it's splattering everywhere as you ride, that might be time to ease up) while failing to keep up with it is the leading cause of dropped chains, broken chains, or just plain hard riding. It's not flexing easily and freely, you're just making more work for yourself.
Oh, and if you have a seized up rusted chain, that you think you might have to replace? Put it in a plastic tup and hose down with a penetrating cleaner like WD-40 or PB-50. Let sit for at least 8 hours. Then take a bucket and fill with soapy water (dishwashing liquid works good) and clean the chain as best you can with a harsh stiff bristled brush. It should already be freed up and working. Take it into the yard and hit with the garden hose using a sprayer to get rid of the soap; then just oil the devil out of it. A good method for this is to let it soak in a pan of motor oil (used works as good as new! Nice recycling tip!) for a few hours, then paper-towel off the excess. Toss it back on ready to ride.
While sure, a replacement chain is only five bucks, a lot of time they won't match the wear pattern of the old sprockets (rear) or chainwheels (front) meaning replacing those too, or facing stiff rides or dropping the chain. It's often better to put the effort into revitalizing/saving the old chain instead of having to replace everything. This wear can also effect gear ratios you don't use a whole lot, which is why many older derailers have certain gear combinations that ride 'harder' than others. You ride around in 6th all the time on a ten speed because you're afraid to shift (know plenty of people who do that too) you're actually making it harder to shift in the long run.
Again, why I like internal gearboxes instead... like my cheap little Shimano Nexus three speed. Everyone thinks it's a single speed, until I drop into third and leave the 22 speeds in the dust... Though I've rarely gotten it up past 20mph. A good comfort speed for me seems to be 16mph. Our friends across the pond in the UK know all about said technology, what with Sturmey-Archer having made them since what? The 1920's or something?
Though I really do think these are the future once they get the weight and price down:
Most 'real' cyclists freak out about how much it weighs, and at $360 just for a bare hub, it's priced out of most 'comfort riders' ballpark.