HOW TO UPDATE SAFARI ON MACPRO 2006

You need to upgrade the OS or switch to a product such as Firefox. To upgrade, choose About this Mac from the Apple menu, check if the computer has at least 2GB of RAM, and if it does, click here and order a download code for Lion 10.7.

Back up your data and check your applications for compatibility before upgrading. In particular, Mac OS X 10.7 and newer don't support PowerPC programs such as versions of Microsoft Office prior to 2008.

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By updating to Lion. Go to the Mac App Store and Purchase Lion and download/install.

Without hacks, your Model Mac Pro can go higher than Lion.

You can try using another browser like Chrome FireFox. Those are the latest even for 10.6.8

I think you meant to say, "can't go higher than Lion."

Yes

If you don't need limited iCloud features, I'd recommend sticking with Snow Leopard and using the latest Firefox or Chrome browsers.