The green berries are grapes, as I hope you realise, they have specific pruning requirements, but I guess you are not asking about those. The white berries are I presume what you are asking about. They something in the Sorbus family, closely related to a rowan. Sorbus with white berries are often S fruticosa or S cashmeriana, but there are other possibilities. But knowing it is a sorbus fairly similar to a rowan is probably enough for you.

You don't normally prune them very much except if you need to reduce it or repair damage. Prune it when dormant in the winter, don't reduce it too much at once, try to keep it balanced, would seem to be sensible.