If you take off into Class B or Class C airspace using VFR, are you automatically cleared into it?

You don't need a clearance into Class C. You do need an explicit clearance into Class B, and it's not automatic, nor is it implied by takeoff clearance. You have to ask for the Class B clearance and receive it before departure. You don't necessarily have to contact Delivery for a VFR flight, but you need to ask for the Class B clearance of whichever ATC you contact.