Instrument is good.

You sing "do re mi fa so la ti" to scales up and down scales, but also sing intervals like do mi so mi do which is a major chord and by singing it you'll better recognise what a major arpeggio sounds like. Likewise, singing do fa do will also teach you to recognise the perfect fourth, which can be confused for a dull perfect fifth for some people.

Also, if you don't want to use solfege names for whatever reason, then numbers work almost as well. Ie one two three four five and then one three five for the arpeggios.

Singing to learn audiation is traditional... I think they stopped forcing singing lessons in the 20th century, so a long, long time ago. In my high school class, which prompted me to self teach rather than rely on teachers, we were told to just know what the intervals are. No explanation how. Just know. Obviously some kids had a pretty severe advantage.