All The Notes On All The Strings
You don't need to memorize this diagram.  But you do need to understand how to find the notes.  To do this, you'll need to know two things:  (1) the open tones of each string, and (2) the notes of the chromatic scale. 

(1). The open tones (the note that sounds when the string is unfretted) of the four strings are G, C, E, and A.  This is the familiar "My Dog Has Fleas" melody that all ukulele players know. 

(2). Each time you move one fret higher on a string, you move one note higher on the chromatic scale.  The twelve notes of the chromatic scale are A, A#/Bb, B, C, C#/Db, D, D#/Eb, E, F, F#/Gb, G, G#/Ab. 

Looking at the diagram, you can see how the chromatic scale extends up the fretboard, one note at a time, starting at the open tone for each string. 

If  you're wondering why the chromatic scale doesn't always have a flat or sharp between each of the "natural" notes of the scale, take a look at the scale as it shows up on a piano keyboard.  The white keys are the natural notes (A, B, C, ...) and the black keys are the sharps and flats.  The irregular pattern of sharps and flats is represented in the irregular pattern of black keys on the piano.  You can see that there are no sharps or flats between B and C, or between E and F.

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