I want to use a new Apple ID on my new phone . Can I backup my old phone to iCloud with an old ID and then restore to my new phone that will have a different ID?

Once you have restored the old backup to the new phone, when changing to the new ID, there may be some data loss, but you can minimize it. If syncing photos with iCloud, ensure Optimization is turned off in Settings/Your Name/iCloud/Photos and/or in Photos/Preferences/iCloud. Then make sure full resolution photos are on the device/computer. When signing out of iCloud, if asked whether you want to retain data, choose yes. Messages conversations will be lost if they are addressed to the ID. See the article below for other places that the old ID should be removed. See Winston Churchill's response in the below discussion. Also set up Family Sharing  between the IDs to keep access to your apps/purchases. 

 How to switch Apple IDs on your iPhone or iPad.      

How to transfer all files from one iCloud account to another.       

What to do after you change your Apple ID or password - Apple Support.      

Family Sharing.         

Family Sharing - Set up.        

Restore your iPhone, iPad or iPod touch from a backup

https://support.apple.com/en-gb/HT204184

No, the restore process will just put your previous AppleID back on the new phone.

If you create a new ID, you will lose any purchase history and stored data attached to that ID. There is no way of merging accounts.