How can I prove $P \rightarrow (Q \rightarrow R)$ is equivalent to $(P \wedge Q) \rightarrow R$
You could just construct the truth tables for the two formulas: there are only three variables, so the tables are only eight lines.
Use de Morgan's laws, distributivity and so on to rearrange one formula into the other.
Use similar techniques to show that $[P\rightarrow (Q\rightarrow R)] \leftrightarrow [(P\wedge Q)\to R]$ is equivalent to true.
Use similar techniques to show that $[P\rightarrow (Q\rightarrow R)] \leftrightarrow \neg[(P\wedge Q)\rightarrow R]$ is equivalent to false.
Use one of the various propositional proof systems to accomplish one of 2&ndash;4, though that sort of thing will probably come up later in your course.