Hi Relfe
I would start with Lanpak, only because otherwise you will get nag messages reminding you to upgrade it. But the order doesn't really matter.
Payables, Ledger and Receivables had database structure changes moving from 8.1 to 9.x. But my guess is that you are already on 9.x for Ledger and Receivables, since they came out so long ago, so the only module that might be an issue is Payables.
Without knowing exactly your modules and what version they are currently, I cannot give you 100% assurance with regards to your developer. That is the kind of thing your Adagio consultant should be able to tell you.
Steve