I have a decent grasp of theory and have seen the value of grinding scales on other instruments. With piano, I'm never sure which fingers I'm supposed to be using or when to move my fingers up the keys. Do you have any guidance?

You know, I can't totally remember how my first teacher taught it (I was 7), but I do know that a lot of the fingering was outlined by CPE Bach back in the 18th century [1]. Just glancing on the Internet (and quickly playing through my scales just now), it seems like I tend to play them closely to this outline: https://pianoguidelessons.com/fingering-scales-on-the-piano/

But, as my current teacher likes to point out, there's no "true" way. He's a bassist, and has mentioned to me a few times some old survey asking pro players to write out their fingerings for some famous passages -- the results were all over the place, as everyone brings their own approach.

[1] https://archive.org/details/BACHCarlPhilippeEmanuel.EssayOnT...

Hi, when you say you don't know which fingers to move do you mean within the context of a piece of music, or just the scales themselves? For the latter, I can give you some advice (though if you search something like "piano scale fingerings" on Google images, you can probably get some fairly standard fingerings for both hands).

Each diatonic (major, minor or modal) scale consists of 7 distinct notes, and the fingering is always 1-2-3 1-2-3-4 in one direction, and the reverse in the other direction, however, you need to find where this sequence starts within the scale. The more black keys there are in a scale, the fewer the possible comfortable positions. Always put your thumb on a white key, and prefer putting your 3rd and especially your 4th fingers on black keys, if possible. (Fun fact: for the major scales, once you have your right hand's fingering, you can imagine mirroring the keyboard and your hand around the D or G# key and you get another major scale with a good fingering for the left hand).

DO NOT start with C major if it's your first time learning scales. Maybe start with E major (4 sharps) as it is comfortable and you can use the mirrored fingering in the other hand.

It's worth picking up a book like Alfred's "The Complete Book of Scales, Chords, Arpeggios & Cadences". It has exercises in every major and minor key, with fingerings.

Knowing the fingerings is a good start, but you need to learn to do these exercises without tension and with proper posture and hand movements. This is something where it's best to have in-person instruction from an expert who can show you the technique and correct your mistakes. But if you can't do that, maybe the next best thing would be to watch some YouTube videos and record yourself playing.

Thumb on white notes. 4th finger on black because it is weaker than 2,3. But these are rules of thumb(!)