@Dannyd Great question. The doctrine of the Trinity, on one level, is a mystery and I think it is healthy to acknowledge that fact. It is clear from Scripture that the Father, Son and Holy Spirit are all God and yet are all distinct Persons. Explaining exactly how that works is likely beyond our abilities. God is the Creator of all things and exists outside of space and time - His essence is far beyond our capacity to evaluate with human reason.
There is a classic misunderstanding of God called modalism that says that the persons of the Trinity represent only three modes or aspects of the divine revelation, not distinct and coexisting persons in the divine nature. I think the danger of comparing the Father, Son and Spirit to son, father and grandfather is collapsing God into one person. One person can be a son, father and grandfather, but God is three Persons. So I think this description does have the idea of roles somewhat right, but is in danger of becoming modalism where there is only one person with three roles rather than three distinct Persons who each fulfill a different role.
@Krams I like the idea of music, but I think that might lead towards partialism?
Partialism taught that Father, Son and Holy Spirit together are components of the one God. This led them to believe that each of the persons of the Trinity is only part God, only becoming fully God when they come together.
@Krams I’ve tried to come up with quite a few analogies myself. I always end up with either modalism or partialism, because you either have three making up one being or one being expressed in three ways. It’s frustrating. Hugh Ross suggested God exists in more dimensions where a single being can have multiple persons. But it’s actually encouraging to me - if God is truly God it is logical to expect He would be beyond understanding in certain ways. Only a man made god would fit neatly within the bounds of our ability to reason. What you would expect from a true God is that His revelation would make the most sense out of reality - origin, meaning, morality, and destiny - but that He Himself would be above our understanding in some ways. And that is exactly what we find in the God of Scripture.