I think basic is on the right track.  The two big classical theorems in analytic number theory whose classical proofs use some complex analysis are [Dirichlet's Theorem on primes in arithmetic progressions][1] and the [Prime Number Theorem][2].  (It is also useful to learn about the combination of the two: the [Prime Number Theorem for Arithmetic Progressions][3].)  

For the former, I can recommend my own lecture notes:

http://math.uga.edu/~pete/4400dirichlet.pdf

http://math.uga.edu/~pete/4400DT.pdf

The second part is explicitly a digested version of the proof Serre presents in his *Course in Arithmetic*.  I don't have a similarly canonical reference to give you for the proof of the Prime Number Theorem (i.e., I don't have any notes on it!), but it can be found in many analytic number theory books, for instance in Apostol's *Introduction to Analytic Number Theory* or, as has been suggested in the comments, Davenport's *Multiplicative Number Theory*.  

[1]: http://en.wikipedia.org/wiki/Dirichlet%27s_theorem_on_arithmetic_progressions

[2]: http://en.wikipedia.org/wiki/Prime_Number_Theorem

[3]: http://en.wikipedia.org/wiki/Prime_Number_Theorem#The_prime_number_theorem_for_arithmetic_progressions