New Pilot here, just planning a VFR trip to KGMU but not sure of the transition rules here. I know flight following will take care of the Class C but who do I need to talk to about the other Class D? Who all do I need to talk too and in what order? Maybe I’m overthinking this one but any help would be great.
On the pilot side the rules are unambiguous. 14 CFR 91.129 (c)(1) requires that you establish two-way communication with "the ATC facility providing air traffic services" before you enter the Class D airspace, unless otherwise authorized in accordance with 91.129 (a). 14 CFR 91.130 (c)(1) requires the same two-way communications prior to entering Class C airspace.
So if you are not on flight following with Greer Approach, you would remain below the Class C shelf (2200' MSL to the south of GSP) and establish contact with Donaldson Tower:
Donaldson Tower, Skyhawk 12345 is ten miles south-southeast at two thousand, transitioning to land Greenville Downtown.
Donaldson Tower will respond with your callsign—which permits you to enter their airspace—and any instructions they might need to issue you. When appropriate, they will either tell you to "Contact Greenville Tower 119.9" or "Frequency change approved." At that point you contact Greenville Tower.
However, as the comments have mentioned, flight following (i.e. radar services from the Greer Approach controller) will be beneficial to you, for three reasons.
First, 91.129 and 91.130 require that you establish communications with "the ATC facility providing air traffic services." This is not always the tower! For example, the hosts of the Opposing Bases podcast work at a Class C approach control that has a satellite Class D tower. They have mentioned that they, the approach controllers, actually own a significant portion of the Class D on the arrival side of the satellite airport—the tower is not the facility providing air traffic services in that part of the airspace, so you would not be fulfilling your 91.129 requirement by contacting them! Another example is the satellite airports in the Boston region; the chart there has a note which says "Aircraft requesting to transit the Class D airspace at or above 2500 MSL contact Boston Approach" even though the surface area goes up to 2600.
Second, if you are talking to a radar controller, the rules they follow tell them to coordinate your surface area transit—or terminate your services—prior to you entering the surface area. There is a Note which says "The pilot is not expected to obtain his/her own authorization through each area when in contact with a radar facility"—but this Note is not, of course, contained in 14 CFR 91.129, so it would still behoove you to confirm that they have coordinated:
Approach, verify Skyhawk 345 may transit the Donaldson Class Delta?
If they say yes, then you may assume that Donaldson Tower—which is "the ATC facility having jurisdiction over the Class D airspace area"—has authorized your being in the airspace despite not establishing communications with them. The authorization is relayed to you by the Approach controller.
Being on flight following also allows you to stay higher for longer because you do not have to avoid the Charlie. This is good because there are some relatively high obstructions in that area, like the 1988' and 1849' towers.