You would back up your data base.You would back up the sites code your running.You would setup a MySQL user on the new server.You would restore the database on the new server.You would restore sites code from the back up you made in step 2.
You may find the following helpful:
the Official Wiki: http://osticket.com/wiki/Upgrade_and_migration
The upgrade guide I wrote: http://www.tmib.net/upgrading-osticket-1812-current-19rc