When you move to different domain, you just need to update URL setting in the admin.
URL to admin will change if you move to new domain.
Example:
[domain_old.com]?s=admin
After you changed domain you access your store admin with new domain and and update setting:
[domain_new.com]?s=admin
If you move to new hosting entirely, you will need to copy all the code and database and do the above.
Hope this helps