What kind of problems (if any) might I have with copyright laws when keeping story notes and some or all of my writing on cloud based storage such as Google drive? How might this affect future publication?
Any writing that you make available to the general public (whether anyone reads it or not) is published. That will prevent you from selling various "first" rights, which is what publishers typically want.
If you haven't made it public, don't worry about it. Google Drive is just storage.
As for the notes... You own the copyright to those, too, but those are not the thing you'll be licensing to publishers. If you make the notes public, and they include substantial chunks of the text of the manuscript, that might cause a problem when you shop the manuscript.
If your notes are mostly ideas for the story, and not the text of the story, that's fine. You can't copyright ideas. Only a particular expression of those ideas.
Google Drive's terms state:
“You retain ownership of any intellectual property rights that you hold in that content. In short, what belongs to you stays yours.”
In general, the form or medium or location in which you keep your writing has nothing to do with your copyright. Whether you write documents with a pencil and keep them in a drawer or save them electronically on your personal computer or on the Internet or chisel them into stone tablets and hide them in a cave, your copyright rights are the same.
If you send your writing to someone with the understanding that they will publish it, of course that's a different category. Then they're not just storing it, they're publishing.
I suppose if you are using any sort of service where someone else has access to your writing in any manner, it makes sense to check any agreements before you sign them.