I'm not a 1Password user so there might be some specific feature I haven't heard about, but from what I understand it works more or less the same as other online password managers
Is there a way to set it up so whatever gets saved to 1Password transfers the login info to my Google account as well?
Most likely not. The idea with password managers is that you're using their website / browser extension / app to manage your passwords, not something else.
You technically should be able to manually export/import passwords between the two, rather than auto sync, but that would be pretty cumbersome to do all the time. It's normally just something people do when initially setting up the password manager (e.g. maybe export your user/passwords from Chrome and then import into 1Password if 1Password has that import feature).
Or should I just download the 1P mobile app, and rely on that exclusively for passwords I need when I’m on my phone?
Yes, that's the general idea with password mangers. You'd install their app and then you would be able to access your synced passwords through there.