Migrating database into a newer version of civicrm?
If you want to migrate the complete database you should indeed do the following:


Set your old environment in read-only maintenance mode
Set up the new environment (CiviCRM &amp; CMS) using exactly the same versions
Migrate the databases
Update CiviCRM and CMS to the latest versions
Properly test (quite some changes between 4.4. and 4.7)
Go-Live or rollback


As an alternative (if your data isn't too complex) you could export everything you need as csv-files and either use the UI import function of CiviCRM to import into the new environment or give Eileen's import extension a shot.