First to answer your question:
The dominant is the 5th scale degree, or the chord built upon this, in any given key. A "fifth" is just a fifth, and that could be between the 3rd scale degree and 7th scale degree for instance. But, neither the 3rd or 7th are called the dominant in relation to the 1st scale degree.
The 7th scale degree could potentially morph into the dominant of the 3rd scale degree, if we have a key change where the 3rd scale degree is now in fact the new 1st scale degree. If this key change is very short in duration, we call it a tonicization, or if its longer in duration, we call it a modulation. The spelling of the chord built upon the 7th degree in fact has changed, and is no longer a diminished triad, but a major triad when we are in this new key.
If you listen to a Bach fugue for example, all of the contrast, departure/arrival, is more or less purely based on key changes (modulation). His music is most often an example of very fast "harmonic rhythm".
The dominant is the most important chord along with the tonic chord. A lot of the other chords are often passed through in harmonic sequences to arrive at the dominant chord. Once at the dominant, you should very quickly, if not immediately, expect the arrival of the tonic. This is called a cadence (and there are sub-categories of cadences, but for all intents and purposes think of 5th to 1st). Ok, sometimes we have what is called the deceptive cadence, usually ending on the chord built upon the 6th degree for instance.
Ok, I don't want to call this "chord built upon 6th degree" for example. I want to call it vi.
So, the scale degrees we will call I, II, III, IV, V, VI, VII. When in a major key, we can use caps or lower case to designate between major and minor (or diminished) triads, so: I, ii, iii, IV, V, vi, vii. vii is actually diminished, but I don't know how to make that tiny circle on the upper right with my computer keyboard to designate it as such.
I will not talk about the dominant 7th at this point in time. However, its movement to I is the most fundamental and common harmonic progression in all of music.
The dominant also often becomes the "new key" in the larger formal sense of many types of works. Now, "Sonata" is one of the most, if not most, ambiguous words there are in music. But, in terms of Beethoven, Mozart, and the like, the nutshell definition could be essentially a passage from tonic to dominant and back home to tonic. (Or relative major perhaps when starting in minor, or even sometimes parallel major, etc). Or in baroque music for example, all of the stylized dance mvmts (assuming major key) in Bach's baroque suites, when using a very typical AABB form, will essentially be tonic-tonic-dominant-dominant.
On the benefits of learning to read music:
You know, one of the easier things about reading music on the keyboard, is that any given note is played only in one darned place. Any same note could be played in multiple different ways on other instruments, OTOH. The biggest benefit of learning to read music is pure efficiency. You get to see everything visually, all at once. It looks crazy at first, but its just like anything else. Matter of familiarity.
On the benefits of learning theory:
To get the basics of "dominant harmony". Essentially, one would should "stress" dissonances with either inflection, volume, etc, and have it resolved during any cadence for example. So, which note of any chord will be particularly inflected? I didn't take the time to talk about dominant 7ths, but that 7th (which you don't know what it is yet, as I haven't explained it) is a particularly good example. That 7th will, 99.9999% of the time, resolve downward, stepwise, to the 3rd built upon the tonic, or I. (i.e. in the key of C major, F resolves down to E). If you don't know this, or at least hear this, then you won't be able to inflect this. I keep saying "inflect" as I can't think of a better word.
One would also want to know if and when they are in a new tonicization or key area. While its up to the artist himself how to inflect the formal aspects of the work, he should at least know when this is happening. Whether small or large scale.
I find romantic works to be easier to understand, generally speaking, in the broad harmonic sense. Usually one is hanging out it in a certain key area for a while. With Bach's incessantly fast harmonic pace, I almost always wrote out the harmonic development on the actual score.
When you can give an overall sense of structure to the piece, it just makes it all more "organic". Even if the audience could not tell you "why" they really liked your performance, they often will. Or hopefully they will!
More random thoughts on classical performance:
Its all in the details. If you played a work exactly the same, except for one most minor improvement (that you did not ignore just to make life easier on your hands), people won't notice. Not any of the non-musicians anyways. Now let's say you make an additional 100 more ultra-subtle improvements for a work of only a 10 minute duration, oh you can bet they will notice. Again, they couldn't tell you exactly why, but they know they enjoyed it a lot. Or hopefully they will!
On repetition:
Practice does not make perfect. Perfect practice makes perfect.
-John Wooden
I hope that helps. You can feel free to ask more questions. On one hand, you might get more than you bargained for, and OTOH, I might say that you should simply take a course in voice-leading/harmony. Depending on the questions asked. IME, the teacher you get is everything. If you were my kid, for example, I'd be asking around like crazy first before enrolling.
If you get a good teacher, intro voice leading is a LOT of fun. Its like sudoku, or tetris, or physics, or chess, but the great thing is you are making music. The first semester will have you able to write stuff that actually sounds good. Like, whoa, I can write for a choir now... I'm not kidding. The problem is it won't be necessarily very original, but it won't sound bad either. It will sound pleasant, with 4, or more, moving voices. In fact, I wouldn't mind retaking a course to refresh myself on some of the more detailed things. Ah well.
Your sentence has been pardoned:
You no longer must suffer meat's endless DS, and now may enjoy your coda. I realize that this might be overkill for someone who briefly worked on the Moonlight. I don't have the score, and I don't think I own Beethoven's sonatas, but going on what I hear in my head:
The understanding of harmonic mvmt, and departure/dissonance towards arrival/consonace could really round out your interpretation on both the small and large scale.
I think* that if I was teaching this work (and I think I'm actually only familiar with the opening bars, and don't know how it goes on, or ends) I would have the chordal accompaniment be steady. I would allow for certain times of rubato, sparingly, but the most important thing to know is which of those chordal tones have the most impact. That creates the foundation and impetus.
The melody can glide over it. Sometimes its hard to do, while keeping the left hand steady. Not to speak of differing personalities/dynamics. I suppose you can record just the left hand, and during playback try out the right hand as lyrically as you can. This is all guesswork on my part, as I am not a pianist, and again I'm unfamiliar with the work. Come to think of it, I would do vice versa as well, as I'm beginning to suspect that its the l.h. that might be much more important than the melody in this work, the Moonlight. I would bet the harmonic inflections are nearly everything in this work, but that's a guess!
Feast on that! hm. Time to refill my bourbon. And fire up the HT?
-jostenmeat
