Sorry for this belated response, interesting question! I cannot quite answer myself how such Greek religious rituals functioned, but I can link some previous discussions on Greek religion in general. I hope you'll get a proper reply from on of our experts soon!

The first thing to consider is that it might be unlikely for your character to identify themselves as a "follower of Hermes". In polytheism it was (and is, to my knowledge) important to worship all the gods, not just choose one. This is explained in detail in the first post of Romanist and military historian Bret Devereaux's blog series "Practical Polytheism" (linked here), which I can really recommend. Directed to (among others) D&D players, he explains the basis of ancient polytheistic religions and why worshipping all gods was important. Then in part two (here) he writes a bit more on how prayers and sacrifices were conducted, which might be more relevant to your question even if he does not go into worship of Hermes specifically. Another post Dr Devereaux has written that could be relevant for the character is this one about oaths and vows to gods. There are also some earlier threads on this subreddit about Greek religion that might interest you, though they are more focused on belief than practice. For instance this thread by u/Iphikrates focuses on how we know Greeks believed in their religion, a thread by u/UndercoverClassicist here discusses how literal belief was, and this discussion by u/MagratMakeTheTea is about how intense personal religious experiences might happen. There is also this answer by u/Alkibiades415 on how religion differed by region that mentions some local variants of deities and festivals in specific areas