First of all, there is no such thing as a Class D airport within Class C airspace. Airspace classes are exclusive; unless there is a cutout in the Class C (which there will be) the C will take precedence over the D.
Answering in the context of faa-regulations, what you should do is call the facility which owns the Class C airspace and say this:
You: Podunk Approach, Skyhawk 123AB, two-zero miles northwest of Podunk at three thousand five hundred, full stop at at Hicksville with information Alfa.
Approach: Skyhawk 123AB, Podunk Approach, radar contact, Podunk altimeter 29.92. Squawk 0101. Maintain VFR at or below three thousand five hundred.
In order to enter the Class C airspace you need to establish two-way communications with the controlling agency, which is Podunk Approach.
In order to enter the Class D airspace you need to establish two-way communications with the controlling agency, which is Hicksville Tower.
At some point Approach will tell you to contact Tower, and you should not switch frequencies before then.
Approach may assign you vectors or altitudes or both to ensure applicable separation between you and other traffic (which in Class C is either target resolution or 500' vertically from IFR traffic, and they will issue traffic advisories and safety alerts on other VFR traffic). It is still your responsibility to see-and-avoid and maintain applicable cloud clearance for the airspace you are in, and you must speak up if a controller's instruction will cause you to violate those clearances.
You do not need to listen to the ATIS for the Class C airport but you should listen to the ATIS for the Class D airport.