Laws of equivalence Need to proof using laws
$$\lnot(p \land \lnot q) \lor q \equiv \lnot p \lor q$$

$\lnot(p \land \lnot q) \lor q$
$\equiv (\lnot p \lor \lnot(\lnot q)) \lor q\quad$    First De Morgan's law
$\equiv (\lnot p \lor q) \lor q \quad$          Double Negation
I think I have the first to laws right. What else do I need to do to prove the equivalence? 

Yes, you've done fine. Now use the associative law to get $$(\lnot p \lor q) \lor q \equiv \lnot p \lor(q \lor q) \equiv \lnot p \lor q$$
You can call the reason for the last equivalence "repetition" (1): $$q \lor q \equiv q\tag{1}$$ $$q \land q \equiv q\tag{2}$$
