Scale Degrees are simply the position of notes within a particular scale.
Check out the following links (also accessible from my sig).
Major
natural minor
harmonic minor
melodic minor
Circle of fourths is a jazz term for the cycle of fifths (a Baroque term).
Both terms are based around the concept of half-close (ii-V) and half-close prepared Perfect Cadences also know as ii-V-I progressions.
Jazz refers to this as the circle of 4ths because bass movement is by moving up a 4th (you can move down the octave within the same chord and then move up a 4th to the next chord).
The Baroque term is about the fact that as you progress around the key centers you are consistently moving up a fifth:
Gb - Db - Ab - Eb - Bb - F - C - G - D - A - E - B- F#/Gb
(again, you can move down the octave within the same chord and then move up a 5th to the next chord).
There are several way to interpret this:
Key based circle of fourths, e.g.
C-F-Bm7b5-Em-Am-Dm-G7-C
I-IV-vii7b5-iii-vi-ii-V7-I
[MP3]http://www.bandcoach.org/audio/cycles1.mp3[/MP3]
or
Am-Dm-G#dim7-CAug-F-Bm7b5-E7-Am
i-iv--vii7b5-III#5-VI-ii7b5-V7-i
[MP3]http://www.bandcoach.org/audio/cycles2.mp3[/MP3]
Major key based circle of fourths
C - F - Bb - Eb - Ab - Db - Gb/F# - B - E - A - D - G - C
I - IV - bVII - bIII - bVI - bII - bV - VII - III - VI - II - V - I
[MP3]http://www.bandcoach.org/audio/cycles3.mp3[/MP3]
It is not necessarily a grand path to hearing chord progressions any more clearly, but it will certainly help with chord recognition in general and movement from chord to chord becomes a little more easy to apprehend as you are listening to new music.