If I made a flyer with some things I wanted to embroider on and sell, and included a picture, do I need to give credit for where the design came from? Or a notice at the bottom that all designs are used by permission? Thanks in advance.

That depends on the designer. With your design you usually get a copyright statement. If not it is usually part of the homepage like here on Cute. I was given a design set as a gift,. It stated that I do not have permission to sell anything using the design.
I am not sure if I answered your question. Pictures from a website belong to the designer as well.