Do you speak Italian?
Personally, I can recognize when Italian is being spoken, and I can usually recognize when it is written, in italics, in my music. Recognizing it is being spoken or written doesn't mean I understand what I am hearing or seeing. I tune out and stop paying attention to things I don't understand, often.
To make it easier for you, I pulled out all of the Italian words in the music and have given you their musical definitions.
Sonata quasi una fantasia
Sonata almost a fantasy
The sonata is written in the style of a fantasia which is a free form and improvisational. This one is almost a fantasia...
This is the first movement of the 3 movements of the Moonlight Sonata. It means to play the piece slowly throughout.
Si deve suonare tutto questo pezzo delicatissimamente e senza sordino
This piece must be played very deicately and without mute
sempre pp (pianissimo) e senza sordino
always very quiet and without mute
growing or swelling- gradually getting louder
gradually getting softer
Attacca subito i seguente
Immediately start the next section without stopping
Without mute is generally interpreted today as "with the pedal.”
According to Charles Rose in his book, Beethoven's Piano Sonatas, a short companion.
"In short, the pedaled sound is still a special effect for Beethoven as it was for Haydn, and he used it above all for contrast. The first movement of the “Moonlight” Sonata is perhaps the only exception in his work, a unique essay in tone color: here he wanted the entire piece to be played with pedal, to be played, in fact, delicately and pianissimo without ever changing the pedal, that is, without lowering the dampers to the strings." (Charles Rosen, Beethoven’s Piano Sonatas: A Short Companion, p. 108)
However, Beethoven's piano was different from the modern ones and the effect of holding the damper pedal down for the entire piece would be really hard to listen to.
Since your goal is probably to play the piece so it sounds nice, experiment. Use the damper pedal (the one all the way to the right) and see what happens. If you put it down at the beginning of the piece and let it stay, how does it sound? What would happen if you changed the pedal (changing the pedal is when you lift your foot off the pedal and put it back on) at the beginning of each measure? How does that sound? Ultimately, the person who's opinion matters most about how the piece sounds is you.