I'm by no means an expert, but I've been homebrewing long enough to have made most of the really bad mistakes...  The kind that ruined beer.  And I've also listened to the mistakes of others...  Because I hate to ruin beer.  So here's a few tidbits of knowledge when it comes to making beer.  Anytime you see something new, you can probably assume that I screwed up a batch of beer.

• Buy Charlie Papazian's Book - The Complete Joy of Homebrewing. It is the Bible of beer making.
• There is an excellent book called Clone Brews.  If you have a favorite beer, a recipe for it may be in this book.  I've used it successfully.
• A weak ammonia solution does a great job of removing labels from beer bottles.  BUT BE CAREFUL.  Never mix Ammonia and Bleach.  It creates a gas that can be fatal.  But...
• Bottling SUX.  Get into kegging as soon as you can.

• Boil or filter all water used for brewing.  If you like the taste of your water, it's probably OK to brew with.  Buy Bottle water if it's not.  Or get one of those fancy filters for the kitchen sink.
• Clean everything... twice.  Wild Yeasts live everywhere, and can easily ruin the perfect recipe.
• Speaking of yeast - use a high quality liquid yeast for the best results.  I highly recommend White Labs yeast. 
• Chill any water that plan to add to the wort.
• Chill the wort as quickly as possible after the boil.  I use an immersion chiller.  If you do small batches, throw your boil pot in a sink full of ice water.  Keep adding ice.
• You can sterilize your immersion chiller by tossing it in the still boiling wort. Just be sure nothing gross is hanging off of it first.

• Bleach is highly corrosive, even for stainless steel.  Use something like B-Brite to clean stubburn stains from the insides of kegs.  Use an Idophor solution to sanitize the insides of kegs and the likes.  Bleach is still fine for cleaning bottles and carboys, but be sure to rinse thoroughly.  Use your nose to be sure the bleach is gone.  If you smell it, keep rinsing.
• Do not add hot wort to an empty carboy.  You stand a very good shot of shattering it.  My sister made this mistake.
• It is better for the wort to be too cold than too hot when adding yeast.  If you add the yeast when the temperature is too high, fermentation might not happen. Thusly therefore, use a Wort Chiller.

• Most brewers use some sort of chiller to get the wort temperature down for yeast pitching, and most of these chillers use water.  Be aware that the water coming out of a wort chiller can be VERY HOT.  And I have the scars to prove it.  Handle your chiller carefully.  In my case, when I turned the water on (via an inline ball valve), the output hose of the chiller did its best impersonation of a runaway firehose.  Despite the fact that I had placed the hose was in a bucket and opened the valve relatively slowly (I thought), the gyser effect was tremendous.  The scalding was quite painful.  Be careful.
• A carboy is fragile and heavy.  Be careful.
• CO2 bottles should not be left in hot cars.

Handy Things:

• The inside of a chest freezer converted to a fermentation chamber will rust because of condensation.  A raised wooden platform will help.  I built mine out of slats off of pallets.  When you see rust, do something about it then.