basic topics are: gcd's, uniqueness of prime factorization of integers, modular arithmetic and fermats little theorem, modular tests for solvability of equations, fermats theorem on sums of two squares, existence of infinitey many primes, refined versions of that result mod various bases.
slightly more advanced topics include gauss's theory of quadratic reciprocity
for studying sqaures in modular arithmetic.
actually this reminds me that one of the first texts on elementary number theory was one of the books of euclid, which is available free online.
the first result is that given two integers, the smallest number that is alinear combination of the two of them equals the largest number that divides both of them evenly.
in euclids language divides is "measures", which makes it clear why the result is true. i.e. notice that the set of all linear combinations of two numbers is a set of equally spaced points on the line, since the difference of any two such numbers is another.
hence the smallest one of these meaures all of them. this is called the gcd.
heres a free link for euclid, see books 7-9 especially:
J. S. Milne's online course notes are nice. If you are planning to study "algebraic number" theory, not just what is usually called elementary number theory, I recommend you to read Milne's "Algebraic Number Theory" available at his website. After that, or while reading it, you may want to read Lang's "Algebraic Number Theory".
If you are also interested in arithmetic algebraic geometry, you will need to study schemes some day. Milne's "Algebraic Geometry" is nice too, but be aware that his notes are written in Serre's language, i.e. schemes without open points. But it is good to bridge classical algebraic geometry and Grothendieck style alg. geo. For this purpose, however, there is a nice book by Mumford called Red Book (LNM Springer).
the OP was asking for free resources. that first book is like a hundred dollars. but it did get good reviews on amazon. i doubt you'll find free resources that will really teach you as thoroughly as a good book anyways.
Most free books that I read, are really bad. Expericence taught me that textbooks are the best way to learn.
The first book which I mentioned, is excellent, it is used by the MIT number theory department.
I still recommend to buy all those books, the pleasure you will recieve is unbelievable. Especially when you first learn the wonderful corellations between field theory with number theory. It is just shockingly beautiful.
I know that free sources are not the best way to go but what can you do when you don't have the money. Everything I make over goes to my college fund or helping my mom pay the bills. That's why I asked for free resources, I'm still finding them to be fairly useful even if textbooks are the better choice.
the elementary algebra notes on my webpage, no. 4, are mostly elementary number theory, and are free. they are class notes from abstratc algebra rather than number theory. a number theorist should also learn quadratic reciprocity for starters, but there is a lot of easy modular stuff there, including fermat's theorem on numbers which are sums of two squares.