If you eat something, does the material have to pass through your digestive system before entering your bloodstream?

Short Yes. By definition, if you put it in your mouth, you just entered the digestive system. That's from mouth to tail.
Yes. That's exactly how it works, in a very simplistic way. It is a complicated process, but that is just how we get nutrients from foods and fluid into our blood stream where those nutrients can be delivered to cells in need.