Sorry if some of this is redundant.
Legend:
# - sharp
## - double sharp
A - augmented
b - flat
bb - double flat
d - diminished
m - minor
M - major
natural
P - perfect
I think the easiest way would be to start by learning diatonic notes and intervals, followed by the major and minor scale, which shouldn't take very long.
There are twelve diatonic notes. A diatonic note always produces the same pitch, regardless of how it's written. These twelve notes make up the chromatic scale, which is what Western classical music, rock, jazz, metal, pretty much everything you've heard is based on.
A
A#, Bb
B
C
C#, Db
D
D#, Eb
E
F
F#, Gb
G
G#, Ab
B to C and E to F are naturally occuring halfsteps, so we don't use accidentals (sharps or flats) between them. However, it's possible to have diatonic notes like B#/C, Cb/B, E#/F and Fb/E, we just usually don't bother. You can also have double flats and double sharps such as Ebb/D, or C##/D, but we use those even less often.
Intervals are counted in halfsteps.
Halfsteps (played sequencially or in a chord):
0 - unison
1 - minor second (m2)
2 - major second (M2)
3 - minor third (tritone) (m3)
4 - major third (M3)
5 - perfect fourth (P4)
6 - diminished fifth / augmented fourth (depending upon where it is in the scale) (d5 / A4)
7 - perfect fifth (P5)
8 - minor sixth (m6)
9 - major sixth (M6)
10 - minor seventh (m7)
11 - major seventh (M7)
12 - octave (P8)
Interval sequences for:
Major Scale - M2, M2, m2, M2, M2, M2, m2
Minor Scale - M2, m2, M2, M2, m2, M2, M2
Tablature for:
Major Scale - 0, 2, 4, 5, 7, 9, 11, 12
Minor Scale - 0, 2, 3, 5, 7, 8, 10, 12
After that, learn the relationships between the two, ie. relative and parallel major.
Relative Major
A Minor - A, B, C, D, E, F, G, A
C Major - C, D, E, F, G, A, B, C
Parallel Major
A Minor - A, B, C, D, E, F, G, A
A Major - A, B, C#, D, E, F#, G#, A
Also notice that the A Major scale follows the same interval sequence as the C Major scale, just starting from a different note. This is called "relative pitch," which means that two different tunes sequentially follow the same pattern of interval leaps although they begin on different root notes. By contrast, "absolute pitch" means that each note has a distinct pitch that never changes, regardless of where it exists relatively within a scale or melody.
After you understand the relationships between relative and parallel major and minor scales, you can do the same with all the other modes, which are as follows:
A - Aeolin Mode (minor scale) * - common in metal
A, B, C, D, E, F, G, A
M2, m2, M2, M2, m2, M2, M2
0, 2, 3, 5, 7, 8, 10, 12
B - Locrian Mode
B, C, D, E, F, G, A, B
m2, M2, M2, m2, M2, M2, M2
0, 1, 3, 5, 6, 8, 10, 12
C - Ionian Mode (major scale)
C, D, E, F, G, A, B, C
M2, M2, m2, M2, M2, M2, m2
0, 2, 4, 5, 7, 9, 11, 12
D - Dorian Mode
D, E, F, G, A, B, C, D
M2, m2, M2, M2, M2, m2, M2
0, 2, 3, 5, 7, 9, 10, 12
E - Phrygian Mode * - common in metal
E, F, G, A, B, C, D, E
m2, M2, M2, M2, m2, M2, M2
0, 1, 3, 5, 7, 8, 10, 12
F - Lydian Mode
F, G, A, B, C, D, E, F
M2, M2, M2, m2, M2, M2, m2
0, 2, 4, 6, 7, 9, 11, 12
G - Mixolydian Mode
G, A, B, C, D, E, F, G
M2, M2, m2, M2, M2, m2, M2
0, 2, 4, 5, 7, 9, 10, 12
As you may have noticed, each mode corresponds to one of the seven white keys on a piano. These are the diatonic modes.
Diatonic means that all seven letters are used once and only once within the scale, either in a natural, sharp, or flat state. Some exotic scales may require double sharps or double flats.
Although there are seven notes, you usually play eight, because you end on the same one that you started on. This is where the word "octave" comes from.
Next, I'd start learning the different modes of harmonic minor. This builds off of your knowledge of relative and parallel scales, so make sure you're comfortable with that stuff before you start learning this.
A Harmonic Minor - A, B, C, D, E, F, G#, A
The only difference is that we have augmented the 7th scale degree. However, we refer to it as "natural 7" rather than sharp or augmented seven, because we describe things in theory by comparing them to the Ionian Mode (major scale) and the harmonic minor scale's 7th degree is a major seventh and therefore a "natural 7."
A - Aeolian Natural 7 (harmonic minor) * - common in metal
A, B, C, D, E, F, G#, A
M2, m2, M2, M2, m2, m3, m2
0, 2, 3, 5, 7, 8, 11, 12
B - Locrian Natural 6
B, C, D, E, F, G#, A, B
m2, M2, M2, m2, m3, m2, M2
0, 1, 3, 5, 6, 9, 10, 12
C - Ionian #5 (harmonic major) [#5 means Sharp 5, not Number 5)
C, D, E, F, G#, A, B, C
M2, M2, m2, m3, m2, M2, m2
0, 2, 4, 5, 8, 9, 11, 12
D - Dorian #4
D, E, F, G#, A, B, C, D
M2, m2, m3, m2, M2, m2, M2
E - Phrygian Natural 3 (commonly called phrygian dominant) * common in metal
E, F, G#, A, B, C, D, E
m2, m3, m2, M2, m2, M2, M2
0, 1, 4, 5, 7, 8, 10, 12
F - Lydian #2
F, G#, A, B, C, D, E, F
m3, m2, M2, m2, M2, M2, m2
0, 3, 4, 6, 7, 9, 11, 12
G# - Altered bb7, Altered o7 (harmonic mixolydian) [o and bb both mean "double flat"]
G#, A, B, C, D, E, F, G#
m2, M2, m2, M2, M2, m2, m3
0, 1, 3, 4, 6, 8, 9, 12
There is also a melodic minor scale that goes like this.
A, B, C, D, E, F#, G#, A
In classical theory, you play it like this ascending (going up), but like a natural minor scale (with natural F and G notes) descending (going down). In Jazz, I think they play the sharp Fs and Gs regardless, but it really doesn't matter. Overall though, this scale is really fucked up and stupid, I wouldn't bother learning it if I were you.
I've reached the limit of my attention span for now though. If I feel like it later, I'll tell you about major, minor, diminished and augmented triads (arpeggios), roman numeral chord progressions, and maybe I'll get into 7th chords.