Tasting (and Troubleshooting)

Tasting and understanding the many flavors of homemade beer comes with experience. It is not much different than tasting various microbrews and learning not only what is proper for a particular style, but what each ingredient does in a beer. Hops provide bitterness, flavor, and aroma that is floral in nature. Different hops have different characteristics and thusly will smell, taste, and bitter differently. Malt is as much the same way a variable as hops. There are different manufacturers and varieties of malt extract such as light, dark, amber, as well as extracts made from specific malt varieties such as Munich malt. There are even specialty grain extracts available if someone wants to experiment with those. This is without even getting into all grain brewing with the infinite combinations there. Taste the bitterness and malt characteristics and see how they balance themselves out in the brew. Take notes. Not only take notes on home brews, but on commercial brews as well. Talk to pro brewers about their brews and if possible, bring a home brew to a pro brewer for evaluation. The best way to learn about beer is by drinking it, especially with others who can provide information on flavor and aroma.
Once the that first brew is bottled, after a few days, it is worth cracking on open for a try.  There is no real way of determining the carbonation time for any particular yeast/beer combination. There should be at least a wisp of carbonation after 4 days (for ales) or so but sometimes it can be fully or mostly carbonated. As the beer (ales specifically) conditions in a bottle, the time before it is fully carbonated the ale will take on characteristics of real ale, the ale on a hand pump or poured from a wooden cask that is not on a draught system.  There have been batches that for whatever reason were not carbonating at all, but that is an extremely rare occurance and something that in most circumstanes should not be a concern. There should be fermentation characteristics in the beer, such as green apple notes, or even a little butterscotch. This is fine and what you want at this bottle fermentation stage.
Different yeasts provide beer with different characteristics. Ale yeasts can be anything from lightly fruity to heavy banana and clove in Belgian brews and German Weiss Beers. Lager yeasts are generally very clean when treated properly and will allow the malt character of the brew to shine through directly. As the beer ferments in the bottle, sulfur aromas will be generated, but these usually get re-absorbed with some conditioning. Training and experience is your friend when evaluating a beer. If the beer tastes good, it is a start. The opinions of others will help refine the beer and encourage the brewer to research the creation and make the needed adjustments. Beer can always be improved, no matter what it is (in most cases anyway).
Clovey/phenolic flavors and aromas or anything that is truly unpleasant generally means that there is a problem. Clove will typically come from a slow start to fermentation, resulting in some baddies that have infiltrated the beer and done a little fermenting themselves. Do not confuse this with some clove character that comes from the fermentation of weiss beer strains and some Belgian strains, usually mixed with banana. Wild yeast strains can be powerful and dominant enough to take over a beer if introduced after regular fermentation is complete (for example, during bottling). Wild yeast will devour the carbohydrates that the beer yeast leaves behind, overcarbonating beer and creating gushers, or worse, exploding bottles (BOTTLE BOMBS!)
Other nasty aromas or flavors include cooked corn or cabbage (often from not boiling long enough), various acidic notes (think vinegar-meaning bacterial infection), and cold cuts (usually caused by yeast autolysis-rotting dead yeast flavor from beer sitting on dead yeast too long). Talk with a homebrewing expert or pro brewer to iron out the problems, or refer to the trouble shooting chart.

Lastly, do not hesitate to indulge in sampling ingredients. Grain should be tasted, hops smelled, and just as important, taste the water. Even compare it to bottled water to assist in spotting problems. Trips to breweries where hop sampling can occur is a must, as breweries have the freshest hops. Even ask if they sell hops to homebrewers. This can boost the quality of homebrew instantly. More importantly get to know what fresh is vs old and stale. Taste fresh malted barley that is used in commercial brews (on your visit). Crystal malt should be sweet and crunchy, not soft or mushy. Roasted barley should have some fresh coffee characteristics. Most importantly and I must reiterate, taste lots of beers as it helps the palate!