Hello henre,
Thank you for using Apple Support Communities.
I understand that you want to get a new Apple ID and begin using that, but you want to keep the information, especially Contacts, from the old Apple ID. It is a good idea to have your own Apple ID in any case. But to make this change like you want, set up the new iPhone, using the backup of the old iPhone and old Apple ID. Once the restore has been completed, you can then use the following support article to sign out of iCloud on your new iPhone.
Sign out of iCloud on your iPhone, iPad, iPod touch, Apple TV, or Mac - Apple Support
You will want to see the section "Sign out on your iPhone, iPad, or iPod touch". Make sure you select the data you want to keep is selected in step 4.
Once you have signed out on your iPhone, you can then use the following article to create your new Apple ID and iCloud.
How to create a new Apple ID - Apple Support
Best regards.