I would memorize the notes perpendicular to the fret board at the following frets, from highest note to lowest note: open strings, 3, 5, 7, 10, 12
- open strings: G, D, A, E
- fret 3: Bb, F, C, G
- fret 5: C ,G ,D, A
- fret 7: D, A, E, B
- fret 10: F, C, G, D
- fret 12: same as zero, or open string
The strings are arranged in fifths. If you learn the circle of fifths you can figure out the notes on those frets.
(Image copyright just plain Bill under Creative Commons license.)
Mnemonic for clockwise:
Fat Cats Give Dogs An Endless Battle (F, C, G, D, A, E, B … Eb, Bb, F)
and counterclockwise:
By Eight All Dates Get Cold Feet (B, E, A D, G, C, F…)
- open strings: G, D, A, E [... Give Dogs An Endless ...]
- fret 3: Bb, F, C, G [Bb (Battle) ... Fat Cats Give ...]
- fret 5: C ,G ,D, A [... Cats Give Dogs An ...]
- fret 7: D, A, E, B [... Dogs An Endless Battle ...]
- fret 10: F, C, G, D [... Fat Cats Give Dogs ...]
- fret 12: same as zero, or open string
Here are a couple of exercises that might help:
Exercise 1:
Find a random note on the fretboard, and think of where it is in relationship to one of those frets. If you don’t see it right away, figure out where the octave is and if you can identify the fret where the octave is, then you can name the note.
Exercise 2:
Pick a note on the low string, like the first fret (F). Then go to the next lowest string and quickly try to find the first F on that string (8th fret). When you find the Fs on all the strings, pick a different random note and find them on each string up to about the 14th or 15th fret.
EDIT: Simpler version of what I wrote above:
F, C, G, D, A, E, B, F#/Gb, Db, Ab, Eb, Bb, F, etc… = Fat Cats Give Dogs An Endless Battle…
When going perpendicular to the strings from high string to low string, the notes will always follow that order, but just 4 notes at a time.