Right off the top I'll recommend a very easy to read and use bicycle maintenance book: Anybody's Bike Book by Tom Cuthbertson. Any good bicycle shop should carry it, in addition to your local library.
For general purposes, we'll be talking about adjustments made to 10-speeds. We'll assume your bicycle is either new, or has had its annual "overhaul" (greasing/adjusting bearings, wheels trued, etc.) at your local bicycle shop.
Most bicycle shops provide a limited-guarantee, free adjustment period (anywhere from 30 days to 6 months). Since bicycle parts have a "wear-in" period (within the first few 100 miles), you should definitely take advantage of this: have them readjust your brakes, derailleurs, chain, wheel hubs, spokes, pedals, headset, and bottom bracket.
You should check the tightness of all nuts and bolts at least once a month, and weekly if you ride daily. (Be careful not to overtighten them as many of the fasteners are made of aluminum and you can strip threads easily.) Check the bolts on handlebar stem and rotation, seat post and mountings, front and rear derailleur mounts, all cable anchors, brake mounts and brake pad nuts, too. (Be sure to check brake pad/wheel rim clearance, which should be about 1/8-inch.) Don't neglect all lever mounts, water bottle holders, racks, fenders, pedals, crank arm and chainring (front gear cogs) bolts, toe clips, mirror mounts, etc.
In performing all of the above, be sure to use the proper tools. Just "making do" with ill-fitting tools can result in minor, but painful, injuries; damaged parts, chipped paint and rounded-off fasteners. A good bicycle shop will have and can recommend all the tools you'll need.
Regularly inspect cable housings and both cable anchors for fraying. If there is ANY sign of wear, replace and lubricate them with grease. Weekly, or better yet, daily, check tires for general wear, and remove imbedded glass or rocks. Tires with large or deep cuts, bulges, worn tread, or sidewall cracking should be replaced. (If you're replacing tires, get ones made with Kevlar, which is a component of bullet-proof vests!)
Check your tire pressure daily. (Correct pressure is stated on the side of each tire.) Proper pressure affords lower tread wear, less tire/wheel damage from unavoidable holes, fewer flats, and lower rolling resistance, making for easier pedaling. To keep everything running smoothly, you should lube your bicycle at least once a month or every 200 miles, and after rain or dusty, sandy, or gritty terrain. Use a spray lube like WD-40, or a lightweight oil. (Keep your wheel rims and brake pad surfaces clean and oil-free by protecting them with cloth, newspaper, and/or paper towels.)
Lube the pivot points of your brakes and derailleurs (and the rear derailleur's "jockey wheels," too), and the chain. Lube the freewheel, being VERY careful not to spray any oil into the nearby rear wheel axle hub, as this could "wash-out" the grease, causing damage. Be sure to wipe off all excess oil when through. As a final touch, give the bicycle a general cleaning and inspect for paint chips and rust; touch-up where needed. Also, use candlewax on exposed cables. With practice, this will all become easy and routine.
A final note: Sealed-bearing components, such as pedals, hubs, bottom brackets, and headsets, require very little maintenance or adjustments, if any, and result in better wear because of their resistance to dirt and water.
