As a direct answer, calculus has no axioms inherent to itself. Theorems of calculus derive from the axioms of the real, rational, integer, and natural number systems, as well as set theory.
Most disciplines of modern mathematics exhibit this sort of behavior, in which the discipline has no axioms inherent to itself. Modern disciplines of mathematics typically work under a unified axiomatic system, the most common one being Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC). ZFC is powerful enough to encode our most frequently encountered structures, including our various number systems.
What, then, do the various disciplines of mathematics working under the axioms of ZFC study? The answer is the various structures that can be constructed using the axioms of ZFC. For example, single-variable calculus can be (very) broadly characterized as the study of real-valued functions with real domain. Such functions can be defined under ZFC, because the real number system can be axiomatized in ZFC.
Similarly, group theorists study algebraic structures called groups, which are defined in ZFC as a set (the building blocks of Zermelo-Fraenkel set theory) and a binary operation on the set, which is a function (which is also defined in ZFC using only sets) obeying certain properties. In a sense, one could interpret these properties as axioms for group theory, but they are actually merely definitions - the underlying axioms are those of ZFC.
The question of truth is a bit trickier to grasp, and there are many differing viewpoints. Complicating the question is that one could adopt an axiomatic system other than ZFC. If you take the perspective that true propositions are those that can be proved in your favorite axiomatic system, then it could easily be that a mathematician that believes in ZFC will disagree on the truth of a proposition with a mathematician that adopts a different system. And different axiomatic systems are not strange or unnatural - indeed, they constitute one of the major objects of study in mathematical logic.
One might also adopt the viewpoint that axioms are not self-evident truths, and therefore one should not believe or disbelieve in any particular axiomatic system. For many of these people, a collection of axioms is a list of meaningless rules to follow, and mathematics is the game of manipulation of symbols under these meaningless rules. (This is Hilbert's formalist perspective.) Truth, then, is a meaningless concept - all propositions are outcomes of some game with some particular rules.
There are other perspectives on truth, many of which I am not familiar with. But this should convince you that the concept of "mathematical truth" is more nuanced than you are probably aware, and really varies from person to person.
As for my personal opinion: I lean toward formalism, so to me it is meaningless to talk about whether calculus "gives me truth." Being inclined toward analysis and topology, I view calculus as an incredibly important and useful tool that agrees with my intuitive understanding of the world, but I view the underlying axioms as just a list of rules I am allowed to play with, without regard to whether I believe in their "truth."
