I am signed into my personal Gmail account through Safari on my phone. If I simply sign into my personal Gmail account on my work computer, can they see any old searches made on my phone or personal computer from home? Or can can they only see what I’m actively searching while on my work computer?
can they see any old searches made on my phone or personal computer from home?
Depending on the network setup, they could inspect your network traffic, get your login details and use it later. That would require you to install a CA certificate on your phone or personal computer, or to log in ignoring your browser warnings about invalid certificates. But logging private credentials can be considered a crime on some jurisdictions, so I doubt they would ever do that. But if they own the network, they can intercept your communications, steal your credentials, log back and see what you have.
But if you haven't installed any CA certificate on your phone or private computer, they cannot. That's exact the point of certificates: prevent any third party to read or change any of the requests you make.
Or can can they only see what I’m actively searching while on my work computer?
Your work computer is different, as they usually have jurisdiction over it. If they have an interception proxy (common on enterprise networks), they can see everything. It does not mean they will see, but they have the capability.