As most platitudinous folk can tell you, an ounce of prevention is worth a pound of cure. Most car breakdowns can be prevented with preventive maintenance. It takes a little extra time and effort, but generally it's worth it to know that your car's not in danger of failing you on the road.
To that end, here's a list of things which you should check regularly. Exactly *how* regularly you need to check them varies. It doesn't hurt to check them every day, of course, but unless you have a lot of time on your hands, that's probably impractical. Generally, once a week should be sufficient.
To check your oil level, wait until the engine has been off for at least ten to fifteen minutes. (If you check it right after the engine has been off, oil will still be pushed up higher than its normal level on the dipstick and you won't be able to read it properly.) Pull out the oil dipstick, and wipe it off on a clean rag. Then put the dipstick back in, leave it for a moment, and pull it back out to check your oil level.
This stuff is really handy if you drive in places where salt, mud, bugs, or other icky things accumulate on the windshield. It's the only stuff which you have any control over the rate of usage, and so you'll probably have some idea of when it needs to be refilled. If you've been using your windshield sprayers a lot lately, you probably should refill the fluid. If not, you don't need to. In case you haven't noticed, there are two kinds of windshield washer fluid: One for summer, and one for winter. The summer stuff is usually pink, and is a pretty mild solution for washing off bugs and bird droppings. The winter stuff is blue, and contains antifreeze (and is thus poisonous) so it doesn't freeze onto the windshield in sub-freezing temperatures.
Unlike oil, which is best checked with the engine cold, automatic transmission fluid is best checked while the engine is not only warm, but also running. When the engine is warm, pull out the transmission dipstick, wipe it off, reinsert it, wait a moment and pull it out again to check your fluid level. The other difference between checking your oil and checking your transmission fluid is how you replenish them: Oil is poured into the oil filler hole, whereas transmission fluid (if you find that you need some more) is poured into the dipstick hole. (Use a funnel for this job, as otherwise you'll spill most of the transmission fluid.)
Take out the air filter and look at it. Generally speaking, if you can't see through it, it's filthy and needs to be replaced. If you don't have the kind of air filter you can see through, use the manufacturer's recommended replacement interval.
Make sure your belts are all in good condition (not cracked or frayed), and at the right tension, not too tight or too loose. Press on the belt to see how much give it has; It should have about half an inch. If there's more, it's too loose, and obviously, if there's less, it's too tight.
Look at all the rubber hoses and the wires under your hood. They shouldn't be cracked, torn, or frayed. If any are, they should be replaced with new ones. (They're cheap, and it's worth it.)
Both of these fluids are usually stored in semi-transparent containers with markings on the side to show you exactly what level the fluids should be at. If either needs topping up, put more in. Some cars have a power-steering fluid container which uses a dipstick; If this is the case, obviously you just check the dipstick instead.
Your windshield wipers should wipe water off your windshield neatly, leaving a clear, streak-free view. If they don't, it's probably time to get them replaced. There's actually a surprising variety of wipers and accessories available, all in the interest of visibility (your most important asset when driving) through your windshield (your most important viewport). There are various lengths and styles of wipers, including dual-blade wipers which actually have two blades running parallel next to each other for better wiping action. Some wiper arms are prone to "wiper lift", a state in which they don't press the wiper blades against the glass properly, but instead "float" above the glass, making them "smear" the glass rather than clearing it, which makes it hard to see. You can get special spring clamps which clip onto the base of the arms to maintain the gentle pressure that's needed on the wiper blades.
It's hard to give an exact estimate on how often you should check your spark plugs, since they wear down relatively slowly. Your car's owner's manual should state how often to replace them, but spark plugs aren't just for making sparks and replacing now and then; They're actually your easiest way to take a look inside your engine, since the spark plugs are the only easily-accessible thing that actually goes into the combustion chamber itself. For example, a quick look at a spark plug will reveal if there's oil leaking into that cylinder (if there is, there will be a coating of oil on the plug). If you suspect trouble with your engine, checking the spark plugs on a regular basis to see what condition they're in might be a good idea.
Check your tire pressure with a tire pressure gauge. If any of them need more air, pump them up to the pressure recommended by the car's manufacturer. (Don't forget your spare tire!) Bear in mind that you should always check your tires when they're cold (when the car has been sitting still for at least three hours). This is because the air inside them heats up quite a bit as the tires are driven on, and of course, warm air expands, so the pressure will read higher. Manufacturer's recommendations for their cars' tire pressures are always cited for cold tires. (If by chance any of them have TOO MUCH pressure, you can let some of the air out by pressing on the little pin in the middle of the tire valve. You'll hear air whooshing out.)
It's hard to tell from inside your car if your headlights or turn signals are burned out, so you should check them regularly to make sure they work. For the turn signals, just turn on your 4-way hazard lights and step out to check them.
The windows of your car, like any other windows, tend to accumulate stuff that obscures them over time, and so need to be cleaned occasionally. This is not something that needs to be done regularly, although most gas stations have squeegees for exactly this purpose, so many people just habitually use them every time they gas up. This may be a bit excessive, but it works at keeping the windows clean. However, it ignores the other side of the windows: The inside. Because they're not exposed to the elements, the insides of your windows shouldn't need cleaning as much, but they may still develop streaks or other undesirable markings. A bit of generic glass cleaning fluid and some paper towel should be good enough to fix this. A word of caution: Do NOT use Kleenex to clean your windows, or any other products designed for use on the human body. Kleenex in particular usually has lotion in it to make it less rough against your nose, and while this is fine for you, it's not good for glass; I used to keep a box of the stuff in my car, and it made a convenient rag when I wanted to wipe off my windshield, so I'd use it for that. Later I kept wondering what the heck this greasy stuff all over my windshield was; A thin film of grease developed on the inside of my windshield, and no matter how much I tried to wipe it away, it seemed to cling to the glass. Finally I realized it was coming from the very stuff I was using to wipe it off. An ironic but educational experience.
This is one of those gray areas: Most manufacturers specify an exact interval of time or distance between timing belt replacements, but you may want or need to replace the belt sooner if it has become damaged, since a damaged timing belt can cause serious damage to some engines if they keep running with it. There's no question that the timing belt can wear out with time, and because of its delicate nature, it needs to be in near-perfect condition. To be on the safe side, you should have it checked at least once a year; Make sure that all the teeth on it are well-formed and that they're not chewed up and that the belt itself isn't stretched or worn anywhere; If it is, replace it.