The Art of Problem Solving (AOPS.org) has a website with abundant links to number theory resources, (guides, exercises, reference recommendations) to begin studying number theory, at varying levels of difficulty - from very basic, to progressively more difficult. E.g., To start, see AOPS's Introductory Topics in Number Theory:
"The following topics make a good introduction to number theory:"
Sieve of Eratosthenes
Greatest common divisors
Least common multiples
Division Theorem (the Division Algorithm)
Simon's Favorite Factoring Trick
(Links available for each topic above.)
Not necessarily directly related to number theory, but more foundational in nature to help prepare you for studying it, see the following
For some background reading in you listed topics: "sets, logic, and reasoning," and to better understand proof-reading/writing (with number theory as examples), you might want to start with
Together, they provide both structural knowledge (the "basics" of "building blocks"), and examples and exercises to get your "mind rolling" and to develop agility in your reasoning.
You can preview each of the books at the links above.