Use logical rules to show that for all propositions p, p → p ∧p
I've been using these rules http://www.mathpath.org/proof/proof.inference.htm
Isn't this just saying that if p, then p and p? Which would mean the same thing as if p then p?
I'm not really sure what exactly I'm supposed to be proving here.
$p \to p \wedge p$ is logically equivalent to $p \to p$ because both are tautologies, but it's a different sentence. If you assume $p$, then you can deduce $p \wedge p$ by the "Conjunction" rule. Finally, eliminate the the assumption to conclude that $p \to p \wedge p$.
No, this sentence doesn't tell you anything you didn't already know. Proving it is intended as an easy exercise in deriving theorems of propositional calculus.