I must place an emphasis on the most common sense aspect of home brewing that is over looked by beginners and even many veterans: Freshness Counts!
Malt Extract: If there is a way to check how old the malt extract you are purchasing is, do it. I have not been an extract brewer for some time, but do use spray dry extract for bottling. I know that there are no dates on the bags so it may be tough. Safest when extract brewing with DME is to buy from places that have a large turnover (lots of people at the store all the time). There should be expiration dates on the malt extract cans so CHECK THE DATE! There are many manufacturers of malt extract out there. As long as it is not old, any may be used. Different manufacturers will provide for different flavors in beer. Feel free to experiment to adjust the flavors to taste. Keep in mind, storage of malt extract should be cool or cold to maintain freshness. If the shop store things warm, the extract available may have been compromised.
Keep in mind that extracts tend to have their own flavor characteristics, and using the lightest extracts available will limit that characteristic in your beer and allow the brewer to adjust the flavor more easily through the use of grains. Brew a beer using only the etract first to find out what it tastes like, then add grains to adjust flavors desired for a batch of beer.
Hops: Hops that come with a kit may need to be checked for freshness (and are usually pellets). Smell them. If they smell of cheese, then go get a fresh supply from the brew shop. Pelletized hops are fine and are easier to deal with than whole leaf hops generally, as they will take up less space in the brew kettle. And always store them in the freezer in an air tite container (they should come from the store in a vacuum sealed package). If you have the opportunity to smell the aroma of very fresh hops (the hops used at your local brewery for example) do so! Every home brewer must learn what fresh hops are all about. When buying hops it is even best to buy by the pound as they will have been handled less. Store them in a jar in the freezer. Whole hops generally may not keep as long as pellet hops so do some quality control before using hops-mainly by learning what fresh is.
Yeast: I would recommend tossing the packet of dry yeast that usually comes with a kit. White Labs and Wyeast both have superior products that will produce a superior beer, though this will need to be purchased separate from a kit. These are dated, and pick yeast that has the furthest out date. The Wyest slap packs may be a beginners best bet here. I always do a starter for the White Labs to make certain the yeast is active, where the Wyest has an expanding pack that is started and watched to make sure it is active before pitching. There will be a section on starters and re-pitching yeast on this site if it is not already up. When using dry yeast, as long as it is not past date, make certain that 10-15 grams are pitched for a proper fermentation start, which is a quantity that is greater than what most kits provide.
Water: Surprisingly enough, this is a very very important component that cannot be overlooked. Chlorine is found in most tap water, but can be removed through the boiling of water. Chloramine is also used in some water sources and it cannot be removed thus rendering said water useless. If there is a way to find out what is used in the local water then do it. Hard water and soft water will also impact your results as they effect both the malt and hop characteristics of the finished beer. I personally use a mix of boiled tap water combined with an equal part of distilled water. This lowers the solids (calcium chloride among others) concentration and makes my beer taste better. Since this is a beginner section, I am not going into details here, but generally speaking, if the water tastes good on its own, it should be fine to brew with.
Specialty Grains: Beer kits for beginners often will include a small quantity of specialty grains. For a Stout it will be some roasted barley. For a red ale it would likely be some crystal malt. For a porter maybe acombination of chocolate malt and black patent malt along with some crystal malt. Using some actual malt will add a fresher malt flavor to the beer. There are numerous recipes available that will allow a brewer to learn the effects of various specialty grains. The rule of thumb is to place the CRUSHED grains you want into a cheesecloth bag and add them to the brew kettle at the start of heating the water, removing when the temperature reaches 170F. This will create a sort of tea with the essence of those grains being in the wort (and beer later). Holding the temperature of the water between 150 and 160 F for 15 minutes is even a good idea for maximizing the effects of the grain on the beer.
Irish Moss: A kit should include a small packet of Irish Moss to add with 15 minutes left in the boil. This is a natural clarifier that will remove a bit of the chill haze in homebrew.