One of the things you cover in a Flight Review is communications with the tower or approach control before entering Class C or D airspace. Unlike Class B where you need to hear the magic words, “Cleared into the Class B”, you only need to “establish communication” with ATC to enter Class C or D airspace. What this means is spelled out in the AIM 3.2.4 Class C Airspace and 3.2.5 Class D Airspace.
Arrival or Through Flight Entry Requirements. Two‐way radio communication must be established with the ATC facility providing ATC services prior to entry and thereafter maintain those communications while in Class C airspace.
1. If the controller responds to a radio call with, “(aircraft callsign) standby,” radio communications have been established and the pilot can enter the Class C airspace.
3. It is important to understand that if the controller responds to the initial radio call without using the aircraft identification, radio communications have not been established and the pilot may not enter the Class C airspace.
Emphasis added. The same verbiage is used for Class D.
This can get a little tricky when you are VFR on flight following of VFR practicing approaches. Early in my flying career I was practicing approaches on a busy day and the approach dropped us right into the traffic pattern. We couldn’t get a word in until we were at the numbers and the tower controller for the Class D told us to leave the Class D immediately. We had expected to be cleared to land and had assumed that our contact with ATC had given us permission to enter the Class D but that was not the case. The controller handling the approach is not the controller providing ATC services to the Class D airspace. The approach controller in Class C is the controller providing services in the Class C airspace so once you have established contact with that controller you do not need to establish contact with the tower to proceed further.
Note that these rules apply to VFR traffic. You don’t have to worry about any airspace on an IFR clearance; you are cleared all the way to your destination when you are given your clearance before takeoff. You still need to pay attention to altitudes, especially when given your departure and approach clearance.