Welcome to the Apple community.
If the contacts on both phones are part of your iCloud account, then yes deleting them on your new phone will delete them on the old one.
If you don't wish to have the contacts on your new phone or wish to have a different set of contacts on your new phone, then disable contact syncing in settings > iCloud. Of course by doing this any contacts you have on your new phone will be on your new phone only and will not sync to any other device.
I recently had my iPhone stolen. I am hoping that by doing the same thing on my new iPhone I will delete my contacts and calendar schedule from my stolen iPhone. I deleted all contacts thru iCloud on my new device then changed my password for iCloud. When I re-signed in on my new phone my contacts came back....and I'm hoping they won't on the stolen phone
Hi Winston C
What if one doesn´t wish to have the contacts of new phone appear on the old one what was stollen?
Same situation happened to me. Already deleted them from Icloud and about to use the icloud find my phone to wipe out all of other inormation from old/stollen device, but
don´t undestrand if when i´ll connect to itunes with new iphone, i´ll be able to getback all the information of the stollen device previously backed up and it if it will re-appear in the stollen device