What books to use to start studying Mathematical Logic? I want to study Mathematical Logic. One concept that confuses me, is that implication is equivalent to '-P or Q'. So, I want to start from the book where this idea first started; but I'm not looking only for this idea, but also other basic ideas of Mathematical Logic.
I guess Boole's Boolean Algebra helped build Mathematical Logic. Can you give a brief explanation of how it and other ideas did (Like the previously mentioned implication definition), where they first started (in which books), and what other classic books talk about them?
 A: What you are referring to with your example is the propositional calculus, also knowns as the sentential calculus, which can be traced back to the ancient Greeks and the Aristotelian logic that emerged from that era. 
Boole's study of Boolean algebras is an algebraization of Aristotelian logic, providing an algebraic model and rules of algebra that faithfully mimic Aristotelian deductions.
Mathematical logic is a very broad subject, of which propositional calculus is a very elementary part of, and is very well understood, and is in fact quite trivial. A large portion of modern mathematical logic is concerned with Model Theory, and to some extent is a study of the expression power of formal languages. 
Related to Boole's boolean algebras are Heyting algebras. These provide an algebraization of a different type of logical system, known as Intuitionism. 
I hope this gives you a very rough idea about what mathematical logic might be, and what it certainly is not. 
A: This addresses mostly your title question.
Before moving on to Mathematical Logic, I'd suggest you get a firm grasp of propositional logic, as Ittay Weiss suggests. You'll want to master predicate logic, and to develop a thorough understanding of quantifiers.  
One very helpful, credible (and freely available!) resource is Paul Teller's Logic Primer. (The link will take you to Paul Teller's website for the Primer, which you can download in pdf format.)  There are two volumes, and together they should provide a firm foundation in first order logic and the basis from which to pursue mathematical logic.
Only then, when you've mastered the fundamentals of formal logic listed above, does it make much sense to dig into Mathematical Logic.
A: "Mathematical Logic", these days, connotes a quite advanced study (it is often the title of third-level logic courses in universities, for example). What the OP means actually wants, I guess, is a lower-level introduction to formal logic (a.k.a. symbolic logic). 
There are a lot of good books out there (and quite a lot of not-quite-so-good ones too!). @amWhy mentions a good one that is now freely available. You'll find a few more suggestions at various levels in the introduction to the Teach Yourself Logic study guide which you can download from http://logicmatters.net/students/tyl/
