They are both correct. Give IT to me > whilst saying this the object that you wish to be given will have been identified already. "Yeah I like that scarf. Give it to me." Give me a chance > you identify the thing you wish to be given in the same sentence. "Give me a book that I can read." "Give me some coins"

Both of them are correct, it is only about the fact what you want to emphasize, GIVE IT TO ME, means to me not to someone else, GIVE ME A CHANCE, is about chance not something else.

Both phrases are correct, in their own right. "Give it to me" is used when you're asking for (usually, requesting for) something that is known/visible/understood by the person that you're directing this statement towards. It may be worthwhile to note that, as general courtesy, prefixing a "Please" to the statement may go a long way in successful completion of the request. "Give me a chance" is usually used in the context of seeking an opportunity to prove something that a person is considered incapable of doing.