You can route any traffic very easy through Tor which offers a much better anonymity. You do NOT always need a VPN. On Android you can use Orbot or Invizible Pro for it.

To access some websites via Tor and not all you can install a second browser on your Android and route only that browser through Tor. In Invizible Pro click on Fast Settings -> Tor Settings -> check that Route All traffic through Tor is off and select the applications which you want to route through Tor. In this case your second browser.