How do I type in a file path?

You can use this shortcut: Ctrl+L.

Or, go to Files and select Enter Location:

See also: Where to configure shortcut keys of Nautilus?

Another method that works is to just click on one of the tabs as if you were going to drill down and just start typing

The area immediately changes to a text box so you can enter a path.

As pointed out by Radu there is a short-cut key for this and you can also get there from the menu. There is a useful list of short-cut keystrokes here:

What are the new Nautilus (3.6) shortcuts in Ubuntu 13.04?

These work in both 13.04 and 13.10.

Edit: In response to comment. Click one of the tabs you don't see any change but if you start typing you see something like this: