(Homework) Prove the law of syllogism Trying to prove, by symbol manipulation, that if $(P \rightarrow Q) \wedge (Q \rightarrow R) \rightarrow (P \rightarrow R)$

I am stuck after doing these steps:

(P $\rightarrow Q) \wedge (Q \rightarrow R) \rightarrow (P \rightarrow R$)

$(\neg P \vee Q) \wedge (\neg Q \vee R) \rightarrow (P \rightarrow R)$

$\neg [ (\neg P \vee Q) \wedge (\neg Q \vee R) ] \vee (\neg P \vee R)$

$[\neg (\neg P \vee Q) \vee \neg (\neg Q \vee R) ] \vee (\neg P \vee R)$

$(P \wedge \neg Q) \vee (Q \wedge \neg R) \vee (\neg P \vee R)$

I am really clueless by this point, this is the first problem propositional calculus problem I ever try. I was thinking maybe if there was some way to negate one of the Qs so I could then group them together then maybe I could get somewhere but I'm not sure how.

Sorry if this is a bad use of LaTex, its my first time using it.
You want to prove that your first formula (expression) is a tautology (has value true, whatever the truth values of the variables $P$, $Q$, $R$).

Use repeatedly the rewrite rule $$A\to B\quad \rightsquigarrow\quad \neg A\vee B \ .$$
The given expression is then equivalent to
$$\neg\bigl((\neg P\vee Q)\wedge(\neg Q\vee R)\bigr)\vee(\neg P \vee R)\ .\tag{1}$$
According to $\neg(A\wedge B)=\neg A\vee\neg B$ and its dual  the left part of expression $(1)$ is equivalent to
$$\neg(\neg P\vee Q)\vee\neg(\neg Q\vee R)=(P\wedge\neg Q)\vee(Q\wedge\neg R)=P\vee\neg R\ .$$
This shows that $(1)$ is equivalent with
$$(P\vee\neg R)\vee(\neg P \vee R)\ ,$$
which is certainly true.
You can prove it using a truth table. A more intuition based "proof" would be to think of them as a stack of dominoes. p is the first domino, q is the second domino and r is the third domino. Since p knocked over (implied) q which knocked over (implied) r, we can say that p knocked over r.

As for a more formal proof I wouldn't know. Maybe try taking 
$[(p \rightarrow q) \wedge (q \rightarrow r)] \rightarrow (p \rightarrow r)$ (which is syllogism in a logical form) and reducing it to a tautology, but that is only a suggestion.

Here is an example using Modus Ponens (Also known as Rule of Detachment)