The bulk of this question seems to be trying to discover the links between these different terms and their subtle differences as applied to anthropology. Discussion of semiotics can be difficult because it is a fairly complex concept applied to what ought to be a very simple practice, i.e., the discovery of meaning in objects, mainly via interpreting their visual significance. The use of the terms "signs" and "signifiers" further complicates the discussion of semiotics. Semantics simply means the relationship or connection between signs and what the signs stand for. A "sign" can be an image, a sound, a concept or idea that represents or refers to an aspect of culture.
To break it down: semiotics is a mode of criticism or discourse that refers to signs and signifiers and their meaning. As one critic put it, "Semiotics tells us things we already know in a language we will never understand." But it need not be this complex. Semiotics can perhaps be most easily applied to the study of images in media: film, television, and advertising. The reason or this is that these forms of visual media often contain cultural contexts that give layers of meaning to the images used.
To use a specific example: in the TV show "Malcolm in the Middle" which is a comedy about a dysfunctional family, there is a scene where the youngest child Dewey (who is 8) is sitting at the table eating cereal. One of his brothers is in trouble and their mother is about to get very angry. This is a situation the boys try to avoid at all costs because they don't to be yelled at or punished. Dewey decides to try and hide his face behind the box of cereal, which looks a bit like the Kellogg's Cornflakes box. This act of hiding behind the cereal box so his mother won't notice him contains a number of semiotic meanings: the "sign" is the cereal box but it holds many layers of meaning. But because this show uses humor to portray a family which is anything but ideal, the idealistic notions portrayed by this sign carry irony and humor. For example, the box can refer to the ideal suburban family who eats a healthy breakfast: a cultural sign referring to the 1950s image of the typical nuclear family. It can refer to a caring mother who feeds her children nutritious food. The use of the box as a fortress to hide behind (Dewey is literally hiding behind a symbol of ideal suburban happiness, quite at odds with his own family situation) gives it another layer of meaning as a signifier. Finally, the cartoon-like drawing of the rooster on the box has layers of meaning as well: the rooster's loud crowing and vibrant feathers convey a loud and flamboyant image, the opposite of what Dewey wants in this moment. The cartoonish nature of the drawing also conveys lighthearted humor, when the humor is of a darker, more menacing tone.
This is one example of how a simple sign, a cereal box, can carry a number of cultural signifiers and meanings.