>>13418822First (the bar) is describing a rule you can use (if you arrived at P and Q, although I'd write them next to each other and not above, then you can arrive at R).
This is a user's rule, so to speak.
This can also be used to specify a grammar.
Second (|-) is making the statement that your formal system you're consideration proves R when P, and Q is given.
This is a proof theoretical claim.
You'll also see |=, which is semantical implication, meaning true in the model. E.g. in a connected graph G with a finite number of vertices, you'll find that every very vertex can be reached from any other. The theory of graphs T, unless you completely restrict the theory, will not prove (|-) that in general, but the particular model G (also a graph) will have that property (|=)
Last is one statement in your logic and would conventionally be read as the claim that R follows from P and Q. But while \land and \implies are part of the vocabulary of the language itself, the "\frac" bar and the "\vdash" turnstile are not.
