Is there a universal keyboard shortcut to go to (focus) the first textbox in a web browser?

When using Firefox, one can set in about:config the value of accessibility.tabfocus to 1.
This will limit when pressing Tab the moving of the focus to text fields only.

In Chrome you could use the extension Input Focus. Press Ctrl+Alt+L to set focus on a page's first text field and then press the combo again to cycle through whatever other text fields are on the page.

The way you put it, I don't think there's any shortcut.

The only way I see is to use Ctrl-L to go to the address bar, and TAB until you get to the text box.