There are FITS libraries for most programming languages (see this list), but FITS is a simple enough format that it should be pretty easy to extract data without the need for such libraries or utilities.
The data portion of an an image is a simple list of pixel values, and of a table is just a catenation of the binary values in each row. The headers are simple enough that it's easy for a person to read the header and see what those rows should be, so my intuition says that it shouldn't be too hard for person to look at the header, and write a short bash script to, say print the rows of a table as an ASCII table that can be processed with awk. After trying it, however, it doesn't seem as easy as I first expected; I generally need to resort to writing (admittedly simple) code.
I'm really looking for something I can run on a system I am unfamiliar with, and will only be working on for a few minutes, such as when I am helping someone else on their computer. So, I'd like to avoid having to install anything, or write anything long or unmemorable enough that would be annoying to write with someone looking over my shoulder.