Can I update to OS X Mavericks if I get a new graphics card?
Which model of Mac Pro do you have? If it really is a 2006 model, that sounds like a MacPro1,1, which isn't supported after Lion (10.7) as it has 32-bit EFI even though its CPU(s) are 64-bit. The fact that you have a Radeon X1900, which was an option on the 2006 model, suggests that you may well be out of luck when it comes to upgrading. 

Mavericks system requirements are here - as you can see, only early 2008 and later Mac Pros are supported. That said, if you're that way inclined, there are ways of getting Mavericks (and Yosemite, in fact) working on unsupported machines - see this guide, for example, or this guide from LowEndMac. 

Basically, you need to use a modified EFI bootloader to boot the newer versions of the operating system. Should any of the earlier links expire, searching for "Tiamo boot.efi" should find an updated link. You will also need to replace your graphics card - newer versions of Mac OS no longer support the X1900.

You can confirm the exact model number by checking About this Mac -> More Info... and then verifying the Model Identifier. Basically

Update 2015-08-3: Add further detail, add Yosemite detail