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.