As you learn your scales, I highly recommend that you also learn to sing them using the "Tonic Sol-Fa" syllables (the stuff of "Do-Re-Mi" from Sound of Music). These serve the same purpose as the numbers, since they are the same in all keys, but are much more singable since there is only one syllable for each pitch (can you sing "seven flat" quickly?)
Using these syllables to sing the melodies that you are learning will help you to recognize common patterns.
The basic syllables (I'll match them up with the scale numbers) are:
I II III IV V VI VII I
Do Re Mi Fa So La Ti Do
C Maj: C D E F G A B C
G Maj: G A B C D E F# G
F Maj: F G A Bb C D E F
ETC.
(When writing out, the syllables are often just abbreviated to their first letter)
Here's some examples with just this much:
"Joy to the World" Do Ti La /So Fa /Mi Re /Do (or D T L S F M R D)
"Away in a Manger" S /S F M /M R D /D T L /S
[Both of these use scales]
"How Firm a Foundation": S L /D L D/ S D R /M R M /S
"Amazing Grace": S L /D M D /M R/ D L /S
"Brethren, We Have Met to Worship": S S L/ D D/ M M D/ F M R D/ S S L/ D D/ M R/ D
[Notice that each of these start out with the same melodic pattern -- S L D]
As you learn your scales, learn the syllables and sing the scale, the intervals from each note (D R D; D M D; etc. then R M R; R F R) and note the qualities (Major, Minor, aug, dim), and sing the triads. (D M S M D; R F L F R; etc.)
Now you ask "What about accidentals (sharps and flats not part of the key)?" Not to worry, the system takes care of that too. It just doesn't take care of #III and #VII or bI and bIV which are extremely rare.
#I #II #IV #V #VI
Di Ri Fi Si Li
bII bIII bV bVI bVII
Rah Meh Seh Leh Teh
Your ascending chromatic scale: Do Di Re Ri Mi Fa Fi So Si La Li Ti Do
1 1# 2 2# 3 4 4# 5 5# 6 6# 7 1
and descending you get: Do Ti Teh La Leh So Seh Fa Mi Meh Re Rah Do
1 7 7b 6 6b 5 5b 4 3 3b 2 2b 1
[When writing them out, often the #'s and b's will be abbrebiated to R# (= Ri) or Tb (= Teh) etc.
Here's some songs to get you started. See if you can figure them out [The slashes and spaces are to give a little sense of the divisions and held notes]:
1. D M S S S /L L S /D M S S S /F M R /
D M S S S /L L S /S M D /R R D
2. M M M M M M M S D R M/ F F F F/ F M M M/ M R R M R S
M M M M M M M S D R M/ F F F F/ F M M M/ S S F R D
3. D R M R M R M/ R M F F M R F/ M F S M S M S/ F S L L S F L/
S D R M F S L /L R M Fi S L T /T M Fi Si L T D D T/ L F T S D
4. M M F S S F M R /D D R M M R R / M M F S S F M R /D D R M R D D/
R R M D /R M F M D/ R M F M R D R S /
M M F S S F M R /D D R M R D D/
I hope that this gives you another idea that may help you in your quest to learn music.