Ramsey theory has had a major impact in Banach space theory. Its first application, it seems, was in the proof of the justly famous Rosenthal $\ell_1$-theorem. Relatively recently, it was used by Gowers in his dichotomy result.
This book seems to contain the major recent results.
Joeseph Diestel's text
Sequences and Series in Banach Spaces contains a nice treatment of some basic results from Ramsey Theorey and several of the applications thereof, including a proof of the Rosenthal $\ell_1$ theorem, to the geometry of Banach spaces.
This article contains many open problems of a Ramsey flavor.
Some other surveys:
1) Ramsey Methods in Banach Spaces, W.T Gowers, contained in chapter 24 of Handbook of the Geometry of Banach Spaces, Vol 2.
2) Applications of Ramsey Theorems to Banach Space Theorey, Edward Odell, 1981, Austin: University of Texas Press.
Of course, none of this is easy. Gowers won the field medal for his dichotomy result. The following quote from his biography seems relevant:
"William Timothy Gowers has provided important contributions to functional analysis, making extensive use of methods from combination theory. These two fields apparently have little to do with each other, and a significant achievement of Gowers has been to combine these fruitfully."
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
Yes, Information Theory is a branch of mathematics, although its practitioners are often found in departments of Electrical and Computer Engineering or Computer Science.