How can I save a text block in visual mode to a file in Vim?

Select the text you wish to save, in either line visual or block visual mode, and

:w new.txt

That's what you type, but you won't actually see exactly what's above. When you press :, you'll go to the command line which will automatically get filled in with selection information. It'll look something like this:

:'<,'>

Just carry on typing the rest (w new.txt) to get

:'<,'>w new.txt

...and press enter.

With the block is selected, you can :'<,'>w other-file, which will write only the selected block to other-file. Hitting : in visual mode should place '<,'> into the command line for you already, so you really only have to type :w other-file.