I want to write an opening for my essay, but I'm not sure which term to use: metonymy or synecdoche. I have a feeling that it is a synecdoche because Google is a part of the Internet, but I would like confirmation.
If you are using 'Google' to stand for all electronic activity, or for the entire net, that is definitely synecdoche.
"Google was invented by Berners."
"Google transforms lives in Africa through mail, education, business, and economies of infrastructure."
If you are using 'Google' to stand for a business using the net that would be metonomy.
There is, as you have realised, much overlap.