Where would one start his investigation into number theory?
One typically starts number theory with elementary number theory, that is one will cover:
Divisibility, Prime Factorization, Modular Arithmetic, Fermat's Little Theorem, Euler's Theorem, Multiplicative Functions, Legendre Symbols and Pythagorean Triples.
The tools required for such a subject is usually a basic familiarity with proof-based mathematics.
However, after that you'll have to study a good deal of analysis or algebra before taking either Analytic Number Theory or Algebraic Number Theory.