The most important single
capacity human beings have is the capacity to reason logically  to
know that from given
premisses, whether believed or not, follows something with necessity if
these premisses are true.
There are several reasons
why this is the most important single capacity human beings have:

Some
reasons why logic is of fundamental human(e) importance


It
is at the foundations of all human inferences, and therewith of human
survival

It
is at the foundations of peaceful, rational and reasonable social
change and all argumentation

Together
with language, logic is what makes human beings really different from
other animals: only human beings can seek peaceful agreement on what
may be the case and what may be done by rational discussion

Without
mathematical logic, computers are hard to think of, design, built or
maintain

Reviewed March
26 2012:
1. There are quite a
few sites dedicated to logic in some sense. Most are maintained by
academics who cater mostly or only to a few fellowacademics in the
same specialism. And while there is nothing wrong with that, it would
be pleasant if I could direct you to a site where you can learn logic
at most levels and enjoy the experience. Unfortunately, sofar I have
neither found that site nor made it myself. (Maybe in the future,
health permitting.)
Those who want to link to
some interesting sites with accessible, amusing, interesting and
civilizing ideas, formulations, games, puzzles relating to
(mathematical) logic and related subjects are advised to check the
following site and the ones after it:
"Factasia is a philosophical fantasy about the future of
society and the future of technology."
and indeed it is, and it contains a lot of logic, philosophy, and many
bookreferences and links. This is very well done, but to delve deeper
in the logic on the site you need more than is on that site. The
bookreferences to do so are there  and I recommend that you download
Mr Jones' site in the zipped version of Factasia he provides for that
purpose, if his site is even a little to your taste, for it is large
and wellorganized, and far easier and cheaper to access once it is on
your hard disk. Also, it'll probably teach you a lot directly or
indirectly if you are in any way seriously interested in philosophy,
logic, computers or mechanical proofs.
At age 15 or so I probably
would have committed murder to be able to read this material. Now you
can do so for free on the internet.
2.
If you really want to understand both the beauty and the use of
mathematics and logic you have to see it applied to all manner of
problems. Here are four pages that contain a great amount of links to
show just this
3. Next, there are some bookreferences to explain what
I mean by "logic". For the moment I list only authors and titles, and
do not know what is in print. All titles except the last recent ones
should be available in any decent university library. Also, when I
reviewed this list in March 2012, it seemed nearly all titles I mention
are available second hand on the internet, while some are still in
print.
Classic expositions

Bertrand Russell

Introduction to
Mathematical Philosophy

Alfred Tarski

Introduction to
Mathematical Logic

Hasenjaeger

Introduction
to the Basic Concepts and Problems of Modern Logic. 
Paul Halmos

Naïve Set Theory

Bochenski

Formale Logik

Good introductions

Evert Beth

Foundations of
Mathematics

Van der Waerden

Algebra

Joseph Shoenfield

Mathematical Logic

Herbert Enderton 
A mathematical introduction to logic 
Geoffry Hunter 
Metalogic 
Marvin Minsky

Computation: Finite
and infinite machines

Good recent books

F.A. Muller

Structures for
everyone

Barwise & Moss

Vicious Circles
(and more: see below)

George Boolos

Logic, Logic and
Logic (and
more)

Raymond Smullyan 
First Order Logic (and more) 
Here a few comments on
these titles  preceded by the general comment that one
fundamental criterion to list them is their clear styles of writing:
This is an exposition of the intuitions and mathematics that went into
Russell's "The Principles of Mathematics" and Whitehead & Russell's
"Principia Mathematica", mostly without symbolism and accessible to
anyone with a clear mind.
In several ways the best introduction to the subject, especially
because it is nonpretentious and clear about fundamentals.
However,
neither Russell's or Tarski's abovementioned texts go far
mathematically (and were not meant to be). One of the best
introductions to the more mathematical side of logic is
This is especially good because he really goes into the intuitions
behind the mathematics, and also contains good expositions of stuff
usually not found in other introductions, while being formally both
rather clear and precise. (It may be that the English title I found and
quote is not quite the same as the German text I read.)
Nearly
all mathematics these days at least uses the notation of settheory and
presumes an understanding of its foundations. Halmos wrote a very clear
introduction, and also wrote several interesting books that treat logic
as a part of algebra: See his Algebraic Logic
for polyadic algebra.
There
is much more to logic than modern mathematical logic. This is the best
history of logic in Western thought I've seen. (There also is a fine
Indian tradition, impressively summarized by  I believe at present six
 volumes of Link to: Karl
H. Potter (I have no idea whether he is
family of Frank Potter above))
Very
wide ranging survey of the subject by a great Dutch mathematical
logician. Subjectwise it is a bit out of date, but stylistically and
conceptually it is not.
Something
similar holds for the next book, that sheds lots of light on
mathematical logic from a mathematical point of view
This book  in fact originally 2 volumes in German  is close in spirit
(but much older) than the expositions in Muller's and Halmos's books
mentioned below. It also is concerned with Algebra in the mathematical
sense, which covers a lot: logic, groups, operators, matrices, fields
etc.
There are many
mathematical expositions of mathematical logic. Shoenfield I found the
clearest. It also covers a lot of material in a fairly small scope.
In some ways the clearest,
simplest and most thorough exposition. Somewhat less fastpaced than
Shoenfield.
Another fine basic
exposition, especially fit for people who did not study mathematics but
who want a mathematically adequate and clear exposition.
This is an excellent very readable introduction to the mathematical
ideas involved in computing (for which you don't need much mathematics:
a clear mind is all that is necessary).
None
of the books I've mentioned sofar has been recently published (or if it
was, like Tarski 's text I mentioned, it is a reprint). The next few
books are recent:
This is the recent doctoral thesis of a Dutch mathematical physicist.
It covers a lot of material, including Quantum Mechanics, but has the
great advantage of being very clear about what theories are supposed to
be. Muller also delves quite deep into the foundations of set theory
and of category theory.
In
general terms, he expounds a version of Bourbaki's structuralist
approach to mathematics based on a version of Ackermann's theory of
sets and classes, using Sneed's, Suppes', and Stegmuller's
structuralist account of what scientific theories are. As the reader
may have gleaned, the general point of view is: Everything  absolutely
everything  is a structure of some kind.
Barwise & Moss
Barwise & Etchemendy

 Language, Proof
and Logic
 Vicious Circles
 The Liar: An Essay in Truth and Circularity

The first of these is an introduction to logic that may have been the
first such book to include computer exercises. I read it long ago,
liked it, and gave it away, so can't say much more about it, except
that I liked it.
The
second is an exposition of paradoxes and vicious circles. It contains a
lot of good clear explanations of recent thinking in mathematical logic
in fields related to this subject including computer programming and
theories of truth.
In
general terms, the authors sketch solutions (or approaches to
solutions) based on the idea to give up one of the standard axioms of
set theory, the Axiom of Foundation, that excludes the existence of
sets that are members of themselves.
This
is also interesting for psychology and philosophy of mind, since so
many issues in these fields involve some kind of selfreference (such
as the one that allows the reader to understand that in this sentence I
am saying something about this sentence and myself using the term "I").
The third is a treatment of the paradox of the
liar that involves a distinction of two kinds of negation. I am
partial to that  kind of  distinction (having thought of it myself in
1975, although I discovered later others did so much earlier) and an
interesting attempted solution of a very tricky problem. For more
attempted solutions (along various lines) see Recent Essays on
Truth and the Liar Paradox, ed. R.L. Martin.
George Boolos

 Logic, Logic and
Logic
 Computability and Logic (with R. Jeffrey)

The first is a recent collection of essays by Boolos. It consists of
articles in three loose groups (whence the thrice repeated "Logic"),
namely about the foundations of set theory (which is the foundation of
mathematics, which is the foundation of everything else  briefly),
about Frege's logical theories, and about various logical subjects,
notably Gödel's theorems and things impossible or impractical in first
order logic.
This
is also interesting for psychology and philosophy of mind, especially
because Boolos discusses higherorder logic (involved in such
statements as: "There are some relations and properties thereof I can
think of you cannot think of  as shown by this sentence, which you,
dear reader, cannot possibly believe to be true") and gives examples of
formulas computers can't compute.
The second is an introduction to logic by
Boolos and Jeffrey that is good and clear and includes an exposition of
Gödel's Theorems and computable functions. I owe the first edition;
there have been later ones (with some corrections of the first and some
new material).
I really should have included Smullyan but forgot to do so in
the original edition of this internet page (that's quite popular, I
found to my pleasant surprise). I can recommend all of his books (I
think: I have read most of them, and what I have read was uniformly
excellent, very readable, and very clear) also those which are not
mathematical or logical. (So, in mitigation I have provided links to
surveys of these books on Wikipedia.)
In
fact, Smullyan published quite a lot of books in three fields, mostly:
Mathematical Logic, Logic Puzzles and Philosophy, though there tends to
be a substantial overlap that consists mostly of logic.
FirstOrder Logic
is a very fine, very clear exposition of propositional and first order
logic including metatheorems (theorems about what systems of logic can
and cannot prove, and/or about consistency of and provability in
logical systems). It is based on a particularly clear version of Beth's
Semantic Tableaux, and includes what is probably the clearest
exposition of the logic of quantifiers.
Diagonalization and
SelfReference: Smullyan got wellknown as a mathematical logician
with his Theory of Formal
Systems, another very clear introduction to the subject of
what formal systems are, precisely, and with work on Gödel's Theorem,
summarized in his Gödel's
Incompleteness Theorems. The book I listed contains versions
of most of the material of these books, and also of another one Recursion Theory for
Metamathematics, and is probably Smullyan's main work in
mathematical logic. It is unlikely you'll find clearer expositions of
the subject, but it should also be said these are genuinely difficult
subjects.
Then again, for those who want to understand Gödel's Theorems
and have a good time, there is (among others):
Forever Undecided
which introduces these theorems and the ideas behind them in the form
of a series of logic puzzles, that also introduce standard logic. This
is listed as one of Smullyan's logic puzzle books on Wikipedia, which
is right in a way  if one realizes Smullyan is the Lewis Carroll of
the 20th Century and all his many puzzle books not only contain very
clever, very amusing and often quite challenging logic puzzles (with
clear solutions!), but in fact are all also introductions to logic.
Another example of this is:
Satan, Cantor and
Infinity: It does consist of logic puzzles, but it is in fact also
an introduction to standard logic and set theory, that also is one of
the most enlightening and amusing introductions to these subjects
(mostly without formalism, but nevertheless quite precise and clear).
Finally, for those really interested in logic:
To Mock a Mockingbird:
This is a book of logic puzzles that also is an introduction to combinatory logic,
which is a foundation for logic and mathematics thought up, created or
developed by Schönfinkel and Curry and later
by others, that manages to derive logic and mathematics from a the
logical combinators Kxy = x and Sxyz = xz(yz). It is quite amazing if
you believed Russell and Whitehead's Principia Mathematica or Zermelo's
Set Theory are what the foundations of lohic and mathematics should
look like or presume. (See also: Lambda calculus.
For an exposition of the relation between these and other subjects, see
Peter Selinger's
"Lecture Notes on the Lambda Calculus".)