Family members getting new phone - want to transfer apps, music, etc to a different Apple ID I don't really use Apple stuff so I'm a little clueless.

My cousin and my uncle currently use the same apple ID. This is problematic because their contacts are synced up, which they don't want.

The idea is for my cousin to get a new iPhone, which will have all of his old music, games, iCloud stuff, etc. but none of his father's contact info.

How is this done?
The big problem in your proposed "ask" is "have all his old iCloud stuff"

What you can do is use one Apple ID with iCloud on two devices. You can even have some of the categories not sync: for example - no contact data, but iCloud has not mechanism to clone or push some content to second account.

You have to manually do that by using two devices and then turning off iCloud on the second device and retain the data you wish on that device. Apple purchases always follow the account that purchased them, so you can possibly mix two "buy" accounts for a short time, but the next time an app update happens, you'll need both account passwords to update everything.

It might be time to make a second account and move forward with each person re-buying / controlling what is contained in each account.

The good news - is contacts is really easy to separate so start there and then you can decide how much work it was to separate those so that each device has their own contacts and then repeat for each "bin" of data that comes with iCloud/Apple ID.