Hello Mary1903,
It sounds like you want to know more about syncing with iCloud. We understand how important this is to you and we are here to help.
iCloud allows your sync all of your devices depending on what settings are enabled. So if you sync your old device and new device it should show the same information on both devices. If you don't want a particular thing sync you would need to disable it in iCloud settings.
More information can be found here: Change your iCloud settings
If the issue persists, can you provide a few more details? What exactly would you want to remove from iCloud that is form your new device? Do you have iCloud enabled on both devices? Also, which iOS versions are you currently using?
Cheers!