Try to use Tor Browser for all of your web surfing. It will offer you far better anonymity than any other browser. Make sure to learn the basics of Tor before using it. If the site you want to visit will not work in Tor Browser, try Firefox intead, but realize these browsers do not anonymize your ip by default.
Tor Browser notes: Using Tor Browser to sign into websites that contain your real ID is counterproductive, and may trip the site's fraud protection. Make sure to check for HTTPS before signing in to a website through Tor. Signing into HTTP websites can result in your ID being captured by a Tor exit node.
Firefox notes: This browser uses Google search by default: replace it with a more private alternative. Iceweasel is a rebranded version of Firefox that does not have to follow Mozilla Trademark Policy.
Why are Chromium, SRWare Iron, et al. not recommended on PRISM Break? More info here.
Warning for mobile devices & Tor: Websites using HTML5 <video> tags will leak <video>-related DNS queries and data transfer outside of Tor.