Damper Pedal
Beginners often over-use the damper pedal. The obvious rule is, if the music doesn't indicate a pedal, don't use it. Some pieces might seem easier to play with the pedal, especially if you start slowly HT, but this is one of the worst traps a rank beginner can fall into that will truly hold back development. The action feels lighter with the damper pedal down because the foot is holding the dampers up instead of the fingers. Thus the action feels heavier when the pedal is released, especially for fast sections. This forms a trap that gradually sucks the beginner into using more damper pedal for fast parts. What these students do not realize is that where pedals are not indicated, it is impossible to play the music correctly at speed if you use the pedal.
Those who use HS practice will rarely fall into this trap because the method gets you up to speed so quickly that you can immediately see that the pedal doesn't belong there. This is another trap that frequently catches students who use the intuitive method. Because they start playing slowly at first, use of the pedal doesn't sound so bad and they get into the habit of practicing with the pedal. Only when they get up to speed, do they realize that the notes are all running into each other and they now have to get rid of a bad, established habit. For Fur Elise, use the pedal only for the large LH broken chords and the one RH arpeggio. Practically all of the two difficult interruptions (except for this arpeggio) should be played without the pedal. Even the parts requiring the pedal should initially be practiced without the pedal until you have basically finished the piece. This will encourage the good habit of keeping the fingers close to the keys and discourage the bad habit of playing with too much jumping and lifting of the hands, and not pressing firmly into the keys.
Coordinating the pedal and hands accurately is not an easy task. Therefore, students who start learning a piece HT with the pedal will invariably end up with terrible pedal habits. The correct procedure is to practice HS first without pedal, then HS with pedal, then HT without pedal, and finally HT with pedal. In this way, you can concentrate on each new element as you introduce it into your playing.
Another point about the pedal is that it must be "played" just as carefully as you play the keys with the fingers. See the references for all the different ways to pedal, when to use them, and how to practice those moves. Make sure that you master all these moves before using the pedal with an actual piece of music. There are some very helpful exercises in the references for practicing proper pedaling. When you do use the pedal, know exactly which move you are using and why. For example, if you want as many sympathetic strings to vibrate as possible, depress the pedal before playing the note. If, on the other hand, you want just one clear note to sustain, press the pedal after playing the note; the longer you delay the pedal, the fewer sympathetic vibrations you will get (clearer note -- see the following section for more detailed explanations). In general, you should get into the habit of depressing the pedal a split second after playing the note. You can get a legato effect without too much blurring by rapidly lifting and depressing the pedal every time the chord changes. It is just as important to know when to lift the pedal as when to press it down.
Inattention to the pedal can slow down technical development much more than many students realize; conversely, attention to the pedal can help technical development by increasing the over-all accuracy of what you are doing. When you do one thing wrong, it becomes difficult to do all the other things right. When the pedal is wrong, you can't even practice the correct finger technique because the music comes out wrong even when the fingers are technically correct.
Most of the HS practice should be done without the pedal, even when the pedal is indicated. While practicing HS, you are only trying to figure out how to move the fingers and manage the passage; you are not trying to make music yet, so the pedal is just an unnecessary interference. The most important reason for not using the pedal at this stage is that technique improves fastest without the pedal because you can hear exactly what you play without interference from previously played notes. Also, the keys feel a little heavier without the pedal, as explained above. This extra workout (without the pedal) makes the playing easier when the pedal is added later.