Potentially any network that you use can be monitoring the addresses that your PC(s) are connecting to. By law, ISPs in many countries have to do this and keep logs to help the authorities protect against crime.
Generally, if you connect to a website using HTTPS (rather than the browser default of HTTP), then there will be end to end encryption so the actual content is hidden although the addresses are not.
A VPN will hide the content of a connection, but not the address of the server you are connected to. A Proxy Server will hie your address from the target site. TOR effectively uses multiple VPNs and Proxy Servers to help hide the connection. Theoretically any of these intermediate servers could be a weakness if they are hacked. Using a VPN or Proxy Server might trigger your ISP or the authorities to look more closely at what you are doing.
Even with VPNs, Proxies and TOR, there is a way for the connection to be tracked from end to end. There has to be or you would not be able to communicate with a server. It can take time, but you could be traced.
Normally the tracing gets most difficult in the last few feet from your local router to the specific computer. Home routers do not log individual connections - they do not have the storage. The authorities can trace back to the specific ISP connection, then they can simply impound every computer in the house and forensically search for evidence.
So, for best security, get your own ISP connection, and don't use the Internet for anything illegal.
I hope this helps.