Here is a rough outline of what we've done so far:

Divisibility
Primes, unique factorization
Greatest common divisors
Euclidean algorithm, linear Diophantine equations
Congruence
Inverses, Euler's totient function (&phi) , Fermat's little theorem, Wilson's theorem, primitive roots, Chinese remainder theorem
Pell equation
Combination law for solutions, existence of fundamental solution, Diophantine approximation
Gaussian integers
Norms, prime and units, factorization of integers in Z[i]
Sums of squares
Rational parametrization of solutions, connections to Z[i].

In the textbook, we've roughly covered chapters 1, 2, 3, 5, 6. There has been some additional material (e.g., greater emphasis on Pythagorean triples geometrically, Euclidean rings, Chinese remainder theorem).