How to upgrade Safari in OS X
OS X automatically looks for updates
OS X will regularly & automatically check for updates to Safari (and any other software you have installed via the App Store) and prompt you to install updates as they are found.
It should prompt you in the top-right hand corner of the screen when updates are found.
If you want to manually check for updates and install them, follow these steps:
Open Software Updates
Click the Apple menu icon in the top left-hand corner of your screen.
The system menu will appear, click the "Software Update..." item.
This starts the App Store application and takes you to the "Updates" section.
The Update screen may look different to the one in the screenshot above. Version numbers shown here are examples and may change depending on what is considered the latest version...
Find and activate the Safari update
On this screen, the App Store will show you all of the updates which are available to you.
It will bundle all of the System Applications (eg iTunes, Safari, Security Updates etc) which have updates available into one main "Software Update" item - shown in the following image.
If you click the UPDATE button, it will install all of those updates in one go. Notice that in some cases this may also require you to restart your computer.
If you click the UPDATE ALL button, it will install all of the System Application updates, as well as any Third Party updates - you may be required to restart with this option as well.
It is generally a good idea to install all of the available updates, but be aware: this may change how some of your software functions in the future. You should always have a backup of your system before you install System Updates.
If you want to upgrade just Safari, click the blue "More" link; this will expand the System Applications, allowing you to pick and choose which updates to install.
An expanded list of software which has available updates:
App Store will now update Safari
The latest version of Safari available for your system will download and install itself. This may take a few minutes depending on the speed of your internet connection.
You can see the blue progress bar showing how much of the installation has occured.
Safari is now up to date
Safari is now shown in the Updates installed in the Last 30 Days section of the App Store Updates.
If you chose to also install other App updates, these should be included there as well.