1-3-5 designates the spelling of a major chord. I-IV-V is a chordal progression.

There is a difference between spelling an individual chord (ie: spelling an individual word) and fitting a chord into a chord progress (ie: fitting a word into a sentence).

It's important to keep these two functions separate.

1-3-5 are taken from the first, third and fifth note of a major scale.

I-IV-V are taken from triads (three note chords) built on each note of a major scale.

In a pure major scale (no alterations of the notes) the I chord will be a major chord. The IV chord will be a major chord. the V chord will be a major chord. The rest will all be minor chords or in the case of the chord built on the seventh note, that's a diminished chord. I-IV-V (major chords and traditionally used more than the others which are called "secondary chords". (Simplified general statement with lots of exceptions particularly in the world of jazz).

This being said, the best way to learn theory is to be able to hear it first. The best way to learn scales in every key is to be able to sing them. That's why every music theory course in school is taught with another along with it called ear training.

If you try to separate music theory from hearing it, it's bound to drive most people nuts. When I have taught it, I always insisted that the student was able to sing the chord, sing the scale, then find them on the instrument. It doesn't matter if you have a great voice or not but the best musicians regardless can sing what they play or write.

Best approach, take a musicianship course with a good teacher or enroll at a local community college.

Frank