I really dislike giveaways that require me to jump through hoops to enter, whether it's on a blog or FB. If I have to like a page/post, share it, post a comment, prove that I shared it, etc. I won't even bother. Just my two cents.
I've seen plenty of giveaways that promote new likes for the page. The page owner will post something like "When this page reaches XXX followers, everyone who likes my page will be entered into a drawing to win a _______!" By doing it that way, they encourage their existing followers to share the page to gain new likes without making them jump through hoops to be entered. It's also a reward for those who've been following the page all along.
Another method I've seen is to post a status asking a question (you basically just want a response with no right or wrong answer). Everyone who responds by a certain time of day is entered into a drawing to win a ____. People are more likely to try to win if it's easy to enter. I've actually won stuff by simply answering a question. :)