Move start date

Move start date

I’m just starting to use Projects and regularly juggle a number of projects simultaneously.  I’d like to move the start date of a project and have the associated milestones, lists and tasks etc updated to reflect the change in start date.
This would be required not just for live projects but also to help forecast resources and timescales for delivering expected or potential projects.  When the project becomes live, the start date needs to be updated and everything associated with it moved to the actual/live start time. 
Can this be done?  Updating the start time doesn’t move any of the tasks???