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So my teacher has his own path for me, which I respect and enjoy. However i feel like we are learning things like scales and arpeggios without actually formally getting introduced to them and run thru them as exercises. Example, one piece of music had a run (blue danube waltz i believe) that went from i believe C > E > C (i think its an "easy" version of an arpeggio or perhaps not a full one). Just to check my understanding of an arpeggio, would that be correct? if i had to guess, the full version (if im even right about this being an easy version) would be C > E > G > C

I don't know the exact definition of an arpeggio.
I would call C > E > G > C an arpeggiated chord, but it is an arpeggio.

At some levels the technical exercises repeat that pattern for four octaves.
Commonly, they include major chords, minor chords, seventh chords, and diminished seventh chords.
I would recommend that discuss it with your teacher before you start practicing them.

I stopped practicing them due to injury, not because arpeggios caused the injury, but because they made things worse.

i don't have an exercises for practicing those specifically. I mean I practice the music and I think the music has them in there which in turn makes me practice them. But nothing like how I do hanon and some various scale work.

Sweet06

Hannon has a section of Arpeggio exercises. My book is at home but I'll look up the exercise numbers and post them tomorrow.