"rental video store" is not correct. "Video rental store" is correct, but more commonly we say "video store" or as Daniel said, the name of the store (Blockbuster, Hollywood Video, etc.).
We usually do not use "borrow" unless we are referring to a library. "Borrow" implies you are getting something for a short time for free, while "renting" implies that you are getting something for a short time and are paying a fee for it.
So we usually say:
"I borrowed a book from the library."
"I rented a film from the video store."
Hope this helps! :)