If I'm a jet aircraft flying IFR, and I climb out from a class D airspace, then inevitably I will get into class A airspace, so, do I need a specific clearance for that, is it all taken care of with me inital IFR clearance on the ground? Because, as far as I know, if you're flying VFR you will need an explicit clearance when entering, for example, class B airspace.

As long as you are operating under ifr, you do not need separate clearances to operate in that kind of airspace. That kind of airspace is meant to keep vfr away from you.