Transfer some data to new Apple ID

Set up the new phone using the old phone and account first.

Read the support article Transfer data from your previous iOS or iPadOS device to your new iPhone or iPad - Apple Support to learn how to transfer data from your previous iOS device to your new iPhone, iPad, or iPod touch. Select one of the three options: Quickstart, iCloud, or iTunes/Finder.

Once the data are transferred with the old account on both phones, sign out of the old account on the new phone and sign in with the new one.

If that isn't working then you need to provide more information about the current setup.

Note that some things cannot be transferred between accounts. While you may find that apps downloaded with the old account will appear on the new iPhone when you have signed in with your old account, once you have the new account on the phone you will have to buy them again when you wish to update them. Look into using Family Sharing between the accounts. To learn how to set up Family Sharing, read this support article --> Set up Family Sharing - Apple Support

In general it is a bad idea to use two different Apple Accounts unless you have a very good reason for doing so.

Apple ID's are not meant to be merged, so this can prove difficult I am afraid