My approach is perhaps more simple than many above.
The word SCALE comes from "Scala", Italian word for ladder.
Just like the rungs of a ladder the notes of a scale, be it Major, Melodic Minor or Harmonic Minor, are at rather precise distances apart.
I hear, and encourage my students to hear, a major scale as two groups of four notes (two "tetrachords"). Each group has the same spacings between notes. Starting from the tonic, Tone - Tone - Semitone.
Take C Major. C D E F and G A B C. The gap between the two groups is a tone, so the whole scale reads T T ST (T) T T ST.
No matter what form of scale or starting (tonic) note I choose I play the first four notes then pause a little before playing the next four so I hear the STRUCTURE of the scale clearly and try to PRE-HEAR the next notes coming towards me. If I am playing a two octave scale I repeat the middle octave before continuing in groups of four.
In the simple scales (few sharps or flats) there are many chances to check intonation against open strings.
I think of scales like flights of stairs that need landings every so often so that you don't lose focus. Incidentally, scales overlap each other so that the first four notes of C Major are the same as the last four of F Major, another reason why I promote listening in groups of four.
Another idea for helping intonation is to practice scales in intervals (3rds, 4ths, 5ths, etc. Especially 3rds). A spinoff from slow careful practice like the above is being so much more confident in keeping a clear mind as you move around the fingerboard locating notes.
Arpeggios are also important. Major, minor, dominant and diminished.
I make worksheets, one for each starting note, of scales and arpeggios over two octaves, starting with bottom E. Each sheet contains Major, Melodic Minor and Harmonic Minor scales. Also Major, minor, Dominant and Diminished arpeggios.
Where possible I work out two good fingerings for each that use different positions (overlaps).
I study two adjacent keys at a time (eg E and F). In this way I pretty well cover all the notes between, using different positions. I have an electronic tuner however this, like the piano, is set to tempered pitch.
To help play in tune with the piano, tune each open string to the same name note played near the center of the keyboard.
Hope all this helps.
DP