The 1 chord, will always be built the root of the key signature of the piece?
Each of the other chords, will generally be constructed with notes from the scale of the key the melody is in, but starting from a different degree? (eg a 2-5-1 is using the 2nd, 5th, 1st degree of the key as the root of the chord)
Hey Digger, yes, that's basically correct. I'm a fan of the 'numerical system' related to scales, chords, & chord tones. For one thing the numbers can easily be transposed to any key. Here are a few conventions to avoid confusion:
Chords are generally referred to using Roman symbols. I, IV, V7, etc. Those indicate the chord root in relation to the key center (ex. In the key of C, the I chord is C, V chord is G, VI chord is A, etc.). And of course, each of those chords represent a collection of notes spelled out in 3rds.
Arabic numerals are assigned to individual notes, either in a scale or for the chord tones in a chord. Scale tones are related to the 'root' of the scale. C maj scale: C=1, D=2, E=3, etc..
Chord tones are related to the root of the chord; G7 chord: G=1, B=3, D=5, F=7 (extensions are 9=A, 11=C, 13=E).
Don't confuse the Arabic numbered notes in a scale with the chord tone numbers. Even though the note G is the '5' in the Cmaj scale, that same G is the '1' in the G7 chord, D is the '2' in the C scale, but '5' in the G chord. The only time the numbers will correspond with the scale tone numbers is in the I chord (C E G B = 1 3 5 7).
That may all be more confusing than clarifying, but once you get used to it, it becomes second nature and is very useful. Hope that helps.