I will do exactly the same thing. I just finished my degree in mathematics but in our department there is not a single course of Number Theory, and since I will start my graduate courses in October I thought it will be a great idea to study Number Theory on my own. So, I asked one of my professors, who is interested in Algebraic Geometry and Number Theory, what would be a textbook that has everything an undergraduate should know about Number Theory before moving on. He told me that A Classical Introduction to Modern Number Theory by Kenneth F. Ireland and Michael Rosen is the perfect choice. He also mentioned that I should definitely study chapters 1-8,10-13 and 17. Another book that he mentioned was A Friendly Introduction to Number Theory by Joseph H. Silverman. He emphasized though that this book is clearly an introduction whereas the previous one gives you all the tools you need in order to study many things that are connected to Number Theory. I hope that this helped you!
Elementary number theory, i.e., modular congruence, linear Diophantine equations, quadratic residues and quadratic reciprocity can be easily studied without experience in higher algebra or calculus, and there a probably several books on the topic accessible to you (in almost every mother tongue). Really, it is a question of taste. One of my favorites is Elementary Number Theory, by Gareth and Josephine Jones. Also, it will cover basic and central concepts of number theory you'll need in any advanced study.
From there, number theory breaks in two major branches (with a great deal of overlap between them, and, of course, not comprehensive branches): analytic and algebraic number theory. For analytic, some calculus may come in hand. As for algebraic, basic higher algebra will be expected.
The standard reference of analytic number theory is Apostol's Introduction to Analytic Number Theory. How much calculus you'll need depends on how deep will you go (it can be single variable, multi variable, complex variable, you'll may also need some general topology, who knows?).
I've never read something specific about algebraic number theory, but the Internet seems to recommend Rosen's & Ireland's A Classical Introduction to Modern Number Theory. From the summary, it appears to cover the basics of algebraic number theory (also, with a lot of overlap with elementary stuff). I've had a good experience with Serre's A Course in Arithmetic, which covers both analytic and algebraic aspects, though it's a very hard book to digest (also, the french original is superior).
From there, you'll probably already have very specific interests from which you will be able to get better references. And for the calculus, algebra and topology, you can find introductory references here on the site. Expect to need at least some knowledge of groups, rings and fields, derivatives, infinite series, topology and complex variables to delve in the most advanced (yet central) areas.
Best Answer
The greatest of all classical books on this subject is An Introduction to the Theory of Numbers, by G. H. Hardy and Edward M. Wright.