Is there a later version of Safari for my OS?

The latest version of Safari is 7 and it's part of Mavericks. It's not available as a separate download for older OS's.

To update Safari you must update your OS.

Yes, 5.1.10 is the highest that can run without updating your OS to something higher than 10.6.8.

See if you can upgrade to Mavericks ..

Mavericks system requirements:

• iMac (Mid-2007 or later)
• MacBook (13-inch Aluminum, Late 2008), (13-inch, Early 2009 or later)
• MacBook Pro (13-inch, Mid-2009 or later),
• MacBook Pro (15-inch or 17-inch, Mid/Late 2007 or later)
• MacBook Air (Late 2008 or later)
• Mac mini (Early 2009 or later)
• Mac Pro (Early 2008 or later)
• Xserve (Early 2009)

Your Mac also needs:

• OS X Mountain Lion, Lion, or Snow Leopard v10.6.8 already installed
• 2 GB or more of memory
• 8 GB or more of available space

From here > OS X Mavericks: System Requirements

Mavericks is free and available for download from the App Store which you should already have access to running v10.6.8.

After Mavericks is installed, you will have Safari 7.0.6 installed. That is the most current version available.

To see how much memory is installed on your Mac, click the Apple  top left in your screen. From the drop down menu click: About This Mac

Please read prior to upgrading > Upgrading to 10.7 and above, don't forget Rosetta! | Apple Support Communities