If you use VPN you can tunnel all traffic including DNS through the tunnel, and then the university won't see what you're doing online. They can see the tunnel though. 

What you search by Google HTTPS should be encrypted, even when not over VPN. The keywords in the URL should appear encrypted in the logs. This is how I understand it, but I haven't tested this so I'm not 100% sure if that is actually the case.

However... organisations can act as MITM (man in the middle) for SSL certificates. This happens and it works and they can do this without you noticing it. They should however mention this in their privacy statement or something. This means they can see your logins to Facebook and Gmail if they want to. If you have some trust in your university, they won't do this in general. If you are a country where human rights are not high on the agenda, they (or the secret service over there) might do this. 

But if you use VPN, that won't work with setting up the connection and the traffic through that tunnel - so I think that is safe.