If you only want to delete select material from your old phone delete the iCloud account (from that device ONLY). Go into Settings > iCloud > Delete Account > Delete > Keep on My iPhone. Following these steps will not affect information on your new phone or your iPad and allow you to delete information from your old iPhone without removing it from your other devices.
If you want to erase all your information from your old phone and sell it or give it to someone else to use you can erase the entire phone and restore it to manufacturer settings. This will allow the new user to set it up under their own Apple ID or restore it from an iCloud back up of their old phone. To do this go into Settings > General > Reset > Erase all Content and Settings >Erase iPhone > Erase iPhone. This method also will not effect your other devices.