Tor (or Orbot) is the safest way to use the web. It uses this special technique to keep you safe; instead of using proxy and VPN, Tor hides you by bouncing through series of PC's all around the world.
As the New York Times writes, “when a communication arrives from Tor, you can never know where or whom it’s from.”
Orbot dosn't require root to work but if you want to enable some extra features like Transparent proxying and some more cool stuff, you might want to grant the #su to this app.
Orbot is available for free on Google Play Store: