Changing your oil is the single most important piece of auto maintenance you can perform. Why does the oil need so much attention? It gets dirty. An internal combustion engine tends to pick up and create a lot of gunk which ends up just sloshing around inside the engine, getting mixed in with the oil. More of it just keeps getting in there as time goes on. Changing the oil is akin to washing out your engine.
So how do you change your oil? Here it is, step by step:
1. Warm up your engine for two or three minutes. This tends to make the oil flow out more easily. (This assumes that your engine hasn't been on in a while and is therefore cold; If that's not the case, just ignore this step. Actually, you should probably let it cool down if it's hot. Changing oil on a hot engine isn't good.)
2. Find the oil drain plug. It's under your car, on the bottom of the engine (specifically, the oil pan). Put a plastic basin under it (to catch the oil as it pours out) and open up the drain with a wrench. Oil should start gushing out. Before opening the drain plug, you may want to take off the oil filler cap from the top of the engine. This really isn't necessary but it does help the oil to flow out better because it eliminates the problem of air pressure keeping the oil inside the engine, just as you can empty a bottle much faster if you cut the bottom off rather than simply opening the cap and turning it upside down.
WARNING: If your engine is hot, your oil will be hot too. This is why it's best to do it on an engine which has been off for several hours, and to only warm up the engine for a couple of minutes. If the oil is hot, don't touch it, of course!
3. When the oil has all run out, unscrew your oil filter, and drain the oil from it into the basin.
3A. Some people recommend starting your engine at this point, and running it for just a few seconds to force the last dregs of the old oil out. This does have some merit to it (there *is* old oil still left in your engine, and running it for a moment *does* help to push that oil out), but other people contend that running your engine at all, even for a few seconds, is not wise when there's no oil inside. Although I've seen people do it without any apparent damage to their engines, it's something you'll want to think twice before doing. Since it's your car, it's your choice, but bear in mind that it's probably not worth the risk of engine damage just to get a few little dribbles of old oil out.
4. Dip your finger in the new oil, and use it to moisten the gasket on the new oil filter. Then put the new filter on. ALWAYS HAND-TIGHTEN THE OIL FILTER. Tightening it with tools will probably lead to over-tightening, and damage to the filter.
5. Replace the oil drain plug, and pour the new oil into the engine.
6. At this point, you're basically done. Start the engine, and take a good, hard look at the oil filter and drain plug to make sure there are no leaks around them. If there are none, check the oil dipstick to make sure you have enough oil. Go driving for a few minutes to get the oil circulating, and check for leaks again. Then check the dipstick again. If everything looks good, wrap it up. Recycle the old oil filter (don't just throw it away, that's bad for the environment), and the old oil too (ditto). Where do you recycle use oil and oil filters? Check your phone book for recycling centers in your area, they'll be able to tell you more. If you have none, ask a service station or auto parts store. Some of them will be able to dispose of it for you in an environmentally-friendly way.