If you use it properly, the Tor browser will protect you against passive surveillance on the local network. It'll be obvious that you're using Tor*, but an attacker won't be able to see the details of what you're doing.
Tor is effectively a more-complicated VPN. All traffic is encrypted as it leaves your computer, and remains encrypted until it reaches an exit node. Someone watching locally won't be able to read the traffic or tell where it's going. Traffic analysis might be able to tell what sort of traffic it is (making a VOIP call, fetching a webpage, and watching a movie all have very different patterns of traffic), but this risk is also present with a traditional VPN.
* Note that this is a risk in it own right. For example, someone who sent a bomb threat over Tor was caught because the police could narrow the source down to a single university, and the person in question was the only one at that university using Tor at the time the threat was sent.