Safari - OSX Updates
To upgrade your Safari browser, you will need to upgrade your OSX to the latest version, El Capitan. The newest version of Safari is part of this bundle. This can take a long time depending on your internet connection, and you won't be able to use the machine while it is upgrading—it's best to plan this update after hours or overnight.
Chrome is not bundled in an OS update like Safari. Click here for Google's own instructional page on updating Chrome.
Below is also a user-created Youtube tutorial video to help illustrate the process of updating Chrome.