The first one is not superset. Well it can be, but in a logical context, it is the horseshoe symbol, and stands for implication. Normally, if you're doing math or something like that, you'll use an arrow, but it's the same thing. The second one is material equivalence. Again, it is a logical connective so it relates sentences only. It is also used to mean other things in other contexts, but in logic, it means that if you have P = Q, then P and Q have the same truth value. The fourth symbol is what some books call "deductive entailment." If you have a set of sentences S and a sentence P, then S |- P means that given a deduction system (i.e. after specifying your logical axioms and rules of inference), you can derive the sentence P from the sentences in S. S |= P is something you might call "semantic entailment." If you're dealing strictly with propositional logic, you might call it "truth-functional entailment" and if you are dealing with predicate logic, you might call it "quantificational entailment." The last two definitions both mean "semantic entailment", but the semantics of propositional logic (logic where you just deal with logic connectives like OR, AND, IMPLIES, NOT, etc.) is different from the semantics of predicate logic (which include quantifiers like "for all x" and predicates). So S |= P means that for any interpretation that each sentence S is true, the sentence P is true.
The single turnstyle "|-" has to do with syntax. It has nothing to do, essentially, with what your sentences mean. Just stipulate a set of rules for manipulating symbols, and if you have some sentences S, where each sentence is just a string of symbols that adhere to some rules as to what counts as a proper string of symbols, then if you manipulate these sentences according to the rules, you can get another string of symbols P, and you can say S |- P. The double turnstyle has to do with semantics. You choose some way of interpreting your strings of symbols. Some symbols have a standard interpretation, like the logical connectives. But if you have a sentence P & Q, then P and Q can be interpreted to mean just about any English sentence you want, but regardless of what you choose them to mean, if P & Q is true, then P will be true, so {P & Q} |= P. So, giving "P & Q" whatever meaning you want with the condition that P stands for some sentence and & stands for "and," but otherwise having total freedom to choose a meaning, whenever your interpretation makes P & Q true, it must make P true.
Note: the first two symbols are logical symbols, the rest (well I'm not sure about "|") are 'metalogical' symbols. So if p and q are sentences of your language, p -> q and p = q are also sentences of your language. p |- q and p |= q are not sentences of your language, they are metalogical sentences about sentences in your language.
I've never seen -||- or =||= before, but I can guess that if you have P -||- Q then it just means P |- Q and Q |- P. That is P is derivable from Q and vice versa. In this case, we say that Q and P are deductively equivalent. Similarly, if you have P =||= Q, then P |= Q and Q |= P, so you would say that P and Q are truth-functionally equiavlent or quantificationally equivalent, depending on the context.
Still not sure about the "|".
