It always surprises me when people think you can buy a bike and then not service it for a year, and wonder why it's creaking and hardly working the next summer.  Then, upon a trip to a local bikes hop, people seemed shocked at the price to repair it, and usually the bike ends up collecting rust in a garage until it gets trashed or donated to a thrift store.  A little basic maintenance can go a long way, and save you a lot of money in tune-ups.  Here's some tips to get you started:

• Get some decent tools - Trying to work on your bike with crappy tools will most likely mess things up more.  Make sure you have a screwdriver set that fits the screws on your bike, and a proper hex wrench set to get you started (I think most bike parts take metric Hex sizes).  As you get to more advanced maintenance, you may need a special cassette tool, cable cuter, chain whip, and bottom bracket tool.  If you want to go all in, and know you'll be doing bike maintenance a lot, you may consider getting a tool kit from a brand like park tool or Pedro's, however a kit still may not have the specific tools you'll need for your bike.
• Start with some basic maintenance - here's some tips, in order of increasing difficulty.
  • Tire Pressure - On my road and mountain bike, ideally i'd check my pressure every ride.  Realistically it's about every other ride.  Tire pressure reduces wear on your tires, and makes riding safer and more enjoyable.  Don't over-inflate or under-inflate! Some roadies believe a myth that over inflated tires reduce rolling resistance, when in reality this actually just makes for a harsher ride.  Mountain bikers will sometimes under-inflate tires for a smoother ride and increased traction, but under-inflated tires are more prone to flats and rim damage.  For an around town bike, you can get by with checking your pressure less frequently, but still check it about once a month!
  • Clean your Chain - Applying lube before you ride, and wiping of the excess, is a great way to make your drivetrain last longer and your bike shift smoother and quieter.  For deeper cleans: I use a grease cleaner from my grocery store called simple green to clean my chain.  I dilute some in a plastic bowl with warm water, and then soak an old t-shirt in it and slowly work each link with the t-shirt until it's not covered in grease. I also will clean off the small gears on the derailer, and try to run the t-shirt between the sprockets on the cassette to remove grease from there.  After the parts dry, be sure to re-lube and wipe of excess lube (excess lube collects dirt and grit!)
  • Know how a derailler works - I'll just tell you: The deraillur is a springloaded device that pushes the chain when you shift gears. When you shift gears, it either increases or decreases the tension in your shifting cable.  If the tension is increased, the deraillur will move one way, forcing the chain onto a diffrent gear.  If you decrease the tension, the spring pulls the derailur back to it's resting posistion and it pushes the chain the other way.
    • Cable Tension - The most common shifting problem occurs when cables stretch out over time, causing the chain to get stuck in-between gears and make a clicking sound.  Most bike's have "barrel adjusters" that can easily add tension to the cable.  Sometimes they're located where ten cable connects to the deraillur, and other times they're located on the shifter.  with the back wheel off the ground, spin the pedals and look for a noisy gear.  Do quarter-turns of the barrel adjuster until the chain seems to shift smoothly and is less noisy.  Don't over tighten the cable, or you could break your shifter.  If it's taking many turns, something else might be wrong (e.g., it may be time for new cables).  If your bike doesn't have barrel adjusters, shift to the smallest gear so the cable has no tension in it, and then loosen the bolt holding the cable and pull some cable through.  In the smallest gear, the cable should have no tension on it, but not hang loose.
    • Limit Screws - it's less likely that these are out of whack, but the limit screws keep the chain from going into the spokes or off the other end of the chain rings.  Deraillurs have high and low limit screws, for when the chain is in the big gear and little gear respectively.  My favorite method of checking the limit screws is to go into the big or little gear and tighten the appropriate limit screw until the gear gets noisy like it's about to shift to the next gear.  Then back off 1 full turn.
    • Deraillur hanger - This can be a real pain, but sometimes your derailleur will get wacked and be bent.  This might be something worth taking into a shop because it could be easy to break your deraliur or worse your frame if you don't fix this right.  if your derailleur doesn't hang vertically, parallel to the gears, chances are, it needs straightened.
  • Brakes - Riding with worn down brakes is not only dangerous because you can't stop as quickly, but can also ruin your rims.  If your brake pads are smooth, chances are their worn down.  Different brakes replace differently. The easiest involve a cartridge that, once a pin is removed, allows the brake pad to be slid out, and a new pad to be slid in.  These require minimal adjustment.  Other brakes require you replace the whole thing.  If you do this, Sheldon Brown (who's website is a mecca of bike maintenance info) has a guide to adjusting diffrent styles of brakes.  If you're replacing a non-cartridge type brake, I'd highly recommend tyring to replace it with a cartridge style to save time in the future.
  • Chain Wear - Aside from regularly cleaning your chain, it also needs be checked for ware.  A stretched chain will shift poorly, and ruin the cassette: requiring you to replace both.  To check for chain stretch, you can buy a special tool or you can just use a ruler.  Put a ruler's zero mark right at the center of a pin on your chain. Now count 12 links (a link consists of both an inner piece and an outer piece, so you'll be counting 24 pins).  The 24th pin should fall right at the 12" mark for a new chain. if you'r chain has stretched, it will be past the 12" mark.  If the pin is between 12 1/16″ and 12 1/8″, you need a new chain.  Past 12 1/8", chances are you need a new chain and cassette.

A great bike maintenance resource is Park Tool's repair guide, which has great information about working on any part of the bike!