get stuck trying to copy or backup database in microsoft SQL server
It doesn't matter if you're running SSMS from your laptop or any other computer.

When you successfully connect to a SQL Server instance using SSMS, you have to get the mindset that you are actually on the server that hosts that SQL Server instance.

So, when you use SSMS to connect to 'OldInstance' (located on Server1) and issue commands like BACKUP DATABASE ... TO DISK D:\MSSQL\BACKUP\backuptest.bak, you are referring to the D-drive on Server1 where that SQL Server instance is located.  If no D-drive exists on Server1, the backup command will fail.  However, your backup completed successfully, so the bak file would be on the D-drive of Server1.

The same idea applies to RESTORE DATABASE FROM DISK ....  'DISK' refers to disks available to that specific server (which could be D-driver, E-Drive, etc. or even a file share).

You don't go into much detail about whether OldInstance and NewInstance are hosted on the same server or not.

Somehow, you'll need to get the bak file that you created against OldInstance to a place where NewInstance can access it.

If 'OldInstance' and 'NewInstance' are on the same physical server (or VM), 
you might be able to work with your hosting provider to help you access the backup created from OldInstance and restore it to NewInstance.

If OldInstance and NewInstance are on completely different servers and they both can't access a given file share, you will probably have to FTP the backup from OldInstance down to your computer (if that is even possible) and the FTP that backup file to the NewInstance server and issue the RESTORE DATABASE on NewInstance pointing the the backup file you just FTP'd