Summarize:

It's the blue app with a white "A" on it; you should see this option in the dock. If you don't see the App Store, click the magnifying glass in the top-right corner of the screen and type "App Store" into the search field, then click the "App Store" result. You'll see this option on the far-right side of the row of options at the top of the App Store window. Doing so will prompt Safari to update to the latest supported version. You can make sure that you're always running the latest version of Safari by checking that automatic updates are enabled: Click the Apple menu and select System Preferences. Click the App Store option in the System Preferences menu. Check the "Automatically check for updates" box. Check the boxes to enable automatic app and system updates.
Open your Mac's App Store. Click the Updates tab. Click Update right of the "Safari" option. Ensure that automatic updates are enabled.