In case nobody else noticed, grounding and bonding are 2 of the most confusing issues that the code book fails to clarify.
Ground rod at generator is optional. I have seen it engineered and passed with a ground rod, and without. You are required to have both a neutral and ground from the generator to the transfer switch.
Neither of the OP's panels were subpanels, but now the panel fed from the transfer switch is.
The non transfered panel can have neutrals and grounds on the same bar, with a main bonding jumper, fed from the meter with SEU cable (2 hots and a ground).
The transfered panel is now a subpanel, neutrals and grounds should be separate, no main bonding jumper, fed from the transfer switch with SER cable (2 hots, neutral, and ground).
The transfer switch (which needs to be service entrance rated in this case), is now the main breaker for the transfered panel, and should have a main bonding jumper, fed from the meter base with SEU cable.