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.
When you are looking at anthropology you are looking at the study of all humans from past to present and how we have progressed not only biologically but socially and culturally.
When we look specifically at semantics in anthropology, we are focusing on language developed within human culture. This is cognitive anthropology, which focuses on the actual meaning behind words within a culture. An example would be how a word denotes a person or action within the language and how this can change over time.
When we look at semiotics in anthropology is similar in terms of language however instead of looking at the meaning of the words you are looking at the symbols that represent them. An example of this is the use of the Rossetta Stone and how, by using it, you can learn the meaning of a symbolized concept by knowing the same concept as symbolized in another language.
When we look at symbolic anthropology, it is different from semiotics in that the specific meaning of the symbol to the culture, not the concept it denotes, is the focus. For example, when you look at Egyptian hieroglyphs one symbol may mean “sun.” When you look at this in the context of the Egyptian culture, what derives from this is symbolic anthropology. If you are looking at a symbol in terms of the concept of the sun as it is applied across cultures, for example Egyptian culture, this is semiotic anthropology.