Prove that there are infinity many tautologies. For this question I think I am suppose to use proof by contradiction, but I need some hints on how to proceed with the proof. Always if someone can give me a brief explanation on how proof by contradiction was that would be really helpful.
Just show a pattern of tautologies that clearly has infinitely many wff's. For example,

$$A\implies A,\ B\implies B,\ C\implies C,\ldots$$

or

$$A\implies A,\ (A\land A)\implies (A\land A),\ (A\land A\land A)\implies (A\land A\land A),\ldots$$

or many other patterns. Make one that you like.
If there exist only finitely many tautologies, then the system {CpCqp} under the rule of substitution and detachment has a shortest member (the length of a wff consists of the number of symbols it has excluding parentheses).  But, we can always substitute any already proved theorem for 'p' in that formula, and substitute any variable for 'q' which does not belong to the already proved theorem.  Thus, we obtain C$\alpha$$\beta$ from $\beta$, where $\beta$ consists of a theorem.  Since $\beta$ represents an arbitrary axiom or theorem (thesis), we can always obtain a longer theorem than $\beta$.  Thus, there is no shortest member of the set of all tautologies, which means we have a contradiction.  So, there exist infinitely many tautologies.