As someone inexperienced in formal music theory, I wonder what the purpose of the first movement of Moonlight Sonata is. Whenever I listen to Moonlight Sonata, I feel like skipping directly past the first movement. What is the significance of this movement? What is the logic behind having this piece of slow and calm music with a few notes? To me it feels unnecessarily dragged out, even if it's meant to create a sense of build up. Why does it fit, and does it form a structured, logical sequence, when connected with the other movements?
The "Moonlight" Sonata is just one of many pieces with the title "Sonata". The title "sonata" is descriptive of how the music is constructed; a basic description can be found in What is “Sonata-Allegro form” and what other similar forms should I be aware of?.
The core idea of a sonata is of musical contrast. For example, @Jomiddnz mentions how the three movements of the piece move from quiet/contemplative, to dance, to dazzling/driven. A more prototypical sonata would have a fast - slow - fast arrangement of movements, and this is one of the ways that Beethoven "breaks the mould" in this particular piece: by arranging the movements slow - medium - fast.
Beethoven himself did not name the piece "Moonlight". Instead, he called it "Sonata quasi una fantasia" (Sonata sort of like a fantasy). A "fantasy" is another type of musical form, but unlike a sonata, which has a more strictly defined set of requirements, a fantasy need not have any particular structure. So again here Beethoven in confounding expectations, primarily in the first movement, by writing in a way that subverts the formal expectations of a sonata and giving it instead a freer, more dreamlike quality. There is some discussion of this aspect in Is Beethoven's title “Sonata quasi una Fantasia” an oxymoron, and does the title instruct how to perform it?
There is a very specific relationship between the first and third movements. The first movement is characterized by the ever-present groups of three notes that undulate beneath the melody. You can hear them particularly clearly in the opening of the piece, but they continue throughout. The third movement opens with the same notes, but transformed into a more aggressive aspect, in extended groups of four notes played much faster. One can try to listen for this by noticing that in the first movement, the three-note groupings are always moving upward. You can hear this similar upward movement in the third movement, but on a grander scale. There is also discussion of the first and last movements in How does the 3rd movement of Moonlight Sonata relate to moonlight?
For an interesting take on the first movement, try Benjamin Zander, for example at(then go back and watch the whole lecture). As you seem to be unimpressed with the usual interpretation, you may prefer Zander's idea, which is somewhat less "dragged out."
The mood of the movement is also rather different at this tempo, being perhaps more turbulent than calm. With that reading, it perhaps forms a better counterbalance to the very turbulent presto agitato of the final movement.
I haven't heard a performance of the whole sonata using this interpretation of the first movement, but it obviously completely changes the proportions. I suspect that you will find the whole to be much more coherent this way. Other answers are of course correct to point out that Beethoven's form is innovative, but that doesn't imply that Beethoven disregarded large-scale form.
You mention in a comment that you don't know the piece very well. In that case, I wonder whether you may just have stumbled on a recording with a particularly boring interpretation of the first movement. Try listening to a few others: it can be done interestingly even at the slow tempo. By "a few" I mean ten or more, sticking to well known professional pianists.
I would like to answer that the reason you want to skip past the first movement to the others is that simply, the first movement has been overplayed, and you are subconsciously bored of it. Typically what comes to mind when Moonlight Sonata is mentioned is neither the second or third movement, but the first movement. Therefore one that is inexperienced in formal music theory probably doesn’t know that a second and third movement exist. After reading Aaron’s fantastic answer, I had to listen to the second and third movement to refresh my memory because it’s been a very long time since I’ve heard them. I honestly forgot how great Beethoven’s masterpiece is. Maybe there is a generalization to Moonlight Sonata’s first movement to act as a gateway to other places that different kinds of music can take you.
I noticed how the third movement finds closure by bringing all three movements together in a grand finale type rendition causing your brain to excrete dopamine letting your body know that you consciously enjoy what you’re hearing. These electro-chemical impulses in your central nervous system(dopamine/serotonin) can create a chemical dependency. So when you want to hear Moonlight Sonata, you’re brain subconsciously wants to have the endorphin release equivalent to hearing the climatic third movement and not the slower introduction to the later pieces. Therefore music can be addictive and habit forming.
Understanding what music does to your brain on a physical level should help you better understand how the music makes you feel. Admitting that the three movements mentioned are brilliant, you fixate on the third, and I don’t blame you.
I tend to personally get bored of music if songs are overplayed, so I’m constantly looking for new material so I can get off so to speak. Good music is hard to find, so when we cannot find anything new that is adequate, we can resort to what we already know is good like Moonlight Sonata. By keeping musical selections fresh prevents collections from getting stale, tiresome and tedious.
I hope this answer helps us realize how much we can enjoy music. Otherwise, it will just be food for thought.
I'll repeat the part of Aaron's answer for emphasis that Beethoven called the Moonlight Sonata a "Sonata quasi una fantasia", which is your warning sign that its movements may not behave conventionally. While Beethoven's previous "Sonata quasi una fantasia" warps multi-movement sonata form even more (slow first movement with faster central section, movements played attacca - i.e. without break), the Moonlight Sonata arguably warps multi-movement sonata form enough with its slow first movement.
The first movement of the Moonlight Sonata behaves like a combination of both first and slow/second movements. Like first movements of sonatas, it is in a sonata-allegro-like form. There is the caveat that, while first movements of sonatas are almost always in pure sonata-allegro form, the first movement of the Moonlight Sonata is in modified sonata-allegro form. Like slow movements of sonatas (which often, but not always, go second), the first movement of the Moonlight Sonata is slow throughout. Funnily, slow movements of sonatas are in modified sonata-allegro form fairly often.
The last two movements of the Moonlight Sonata have the same tonic as the first movement and fit the usual conventions for the last two movements of a 4-movement sonata (triple-meter dance movement in compound ternary form, fast last movement in sonata-allegro, rondo, sonata-rondo, or theme and variations form), thus forcing at least a tonic- and structure-based fit with the first movement.
The first movement exemplifies Beethoven's ability to break the mould and be original. I am sure that one day you will appreciate this movement's gravitas and originality. It creates an impressive atmosphere with a scarcity of material. Then follows a much lighter sounding minuet type movement to relieve the tension, and lastly a dazzling and driven final movement to balance the first two movements.
It depends on how you hear it or what you get reminded of when you hear it. As I am not a professional, I cannot give a logical reason for this observation, but while listening to the whole piece, one is reminded of the phases of life: one being sad, the other happy, and so on. The first movement is calm and makes one think about things that give comfort, but yet a feeling of pain. It's just Beethoven's talent. I advise you: listen to the first stanza and think about the excellence of the piece. Where does the melody take you? I think that might help you accept the piece more readily.