Many students I have worked with have a period of time when they are trying to physically get comfortable with their instrument. Whether it is piano keys, guitar strings, or clarinet/trumpet/etc. keys there are ways students have to use their hands to maximize their comfort and relaxation.
I tell students many times that their fingers should be doing natural things on their instruments, and if things feel tense or uncomfortable, we need to figure it out and change it! There are several reasons for this:
Each instrument has its own challenges and unique combination of stretching and relaxing of the hand.
For the piano, most things I notice is the hand "fighting itself" with some fingers curved and relaxed and some fingers not. Mostly the 4th and 5th fingers are flat and fingers 1, 2 and 3 are looking good. To help correct this, I have the student pick up a squishy ball and hold it lightly with all 5 fingers. Then I have them pick it up with thumb and 2nd finger, thumb and 3rd finger, etc. If they are having problems co-ordinating several fingers together in a chord I have them pick up the ball with whichever fingers are playing the chord. This gets them thinking about what it feels like for them to use their hand in the "normal" situation of picking up an object, and be able to use that feeling on the piano to stay more relaxed.
The guitar has each hand doing a specialized activity, but most tension arises in the left hand as it is working to press the frets and get a clear sound from the string. There is a similar idea here of having the hand act like it would normally act when picking up an object, but there is more stretching and squeezing involved in the activity of pressing. For working on this I have the student pick up the squishy ball with each finger and thumb and have them squeeze a little bit. Usually there will be a lot of unnecessary tension in the hand as muscles are being used besides just the ones for that individual finger/thumb combination, and we work to relax that.
For valve wind instruments (trumpet, baritone, tuba) the main goal is to make sure the instrument is being held/balanced by the hand that is not operating the valves. That way the valve hand can stay relaxed and open because it is not holding onto the instrument. The thumb should find a spot so that the fingers can push against it in a natural way. Once again using the squishy ball to press each finger against the thumb can help with this co-ordination.
For keyed wind instruments (clarinet, flute, saxophone) there is a very unique combination of holding, stretching and relaxing the fingers. Both hands are holding the instrument and operating the keys, so isolating the precise muscles for each fingering combination can be a long process. To help the fingers feel how to stretch I often use a rubber band to loop on 2 adjacent fingers and have them do a resistance stretch with it. That way the students can feel how to make space in their hand and give those fingers some extra reaching power.
There are many other individual hand techniques, but these are the main ones that come to mind.