Study the score at hand. The piece is in C-sharp minor. Locate the four sharps (F sharp, C sharp, G sharp and D sharp) and commit them to memory.
Count the beats to yourself. Note that the time signature is "C," which stands for common time. It is a symbol for 4/4, meaning there are four beats in each measure and four quarter notes to count as one beat. However, the piece is written in triplet configuration, so there are three notes to one beat.
Warm up your fingers first by playing some arpeggio exercises, since your fingers will constantly reach octave-wide intervals in the sonata. Begin by playing the left hand and right hand separately. Play them together when you feel confident enough.
Play as far as 16 measures, or the first page of the score. Repeat the process until you master the notes without mistakes. Move on to the rest of the score 16 measures at a time. It is easier and less discouraging if you break down the piece into sizable chunks.
Play the piece from start to finish. Note any mistakes you make by marking the score with a pencil. Go back and target those problem areas by playing just those measures. Write down the fingering at the difficult areas.
Use a metronome to time your playing. Adagio sostenuto means "slow" and "sustained." If your printed copy does not include metronome markings, just set the metronome to a speed that you feel is right for the piece. Stop using a metronome when you feel you have achieved an even tempo.
Study the musical notations, abbreviations and pedal markings in the score and apply them to your performance. Listen to recordings of Moonlight Sonata for reference. Do not mimic the style of other performers. You will soon develop your own interpretation and your emotion will come through in your performance.