If you live in Google's world, check out Chrome. You can use Google's browser across your desktop and mobile devices, syncing bookmarks, passwords, and apps. Chrome comes with a rich collection of Google and third-party extensionsfor making your browsing experience easier. It also has an incognito mode that won't save browsing history, cookies, and information entered in forms. (Websites, your work or school, and your ISP, however, may still be able to track you.) Google updates Chrome constantly but it can feel a bit of a resource hog.
For Mac-focused users, it is hard to do better than Safari, with its Apple-centered approach to Web browsing. You sync passwords, bookmarks, history, and tabs across devices through iCloud. Tap the Apple Pay button on a commerce site to make a purchase. And you can hand off webpages between Mac and iOS devices. It's only available on Apple devices, so if you have a Windows or Android device in your life, you'll need to consider mixing and matching browsers.
Firefox fills an interesting spot on the list of Mac browsers. It's the spunky open source browser not controlled by industry heavyweights, promising to protect your privacy. And it comes with a broad collection of extensions and is highly customizable. But its features overlap quite a bit with Chrome and Safari -- you can sync your settings across Firefox on different platforms, for example, just like you can with Chrome and Safari. And Mozilla is constantly working to speed up Firefox. If you want to steer clear of Apple and Google, then Firefox is the way to go.