What is the purpose of having obvious axioms in mathematics? Why not simply use right away? Why is an axiom and what is not? 
Some of the obvious statements are made into axioms whereas other obvious statements aren't.
For example, the axiom of choice, arbitrary union axiom, etc. Isn't it so obvious?
 A: In general, "obvious" statements sometimes turn out to be false. For example, it was assumed for a long time that any continuous function must be differentiable except at isolated points, until a counterexample was constructed by Weierstrass. There are many other instances of this throughout history.
So if you want to make sure a statement is true, no matter how obvious it seems, you either need to prove it as a result of things you already know, or you need to take it as an axiom. What should be axioms and what should be theorems is a subtle question with no universal right answer, but in particular, the axiom of choice cannot be proven from the ZF axioms (the most commonly-used set theoretical foundation), so if you want to use it, you have to take it as an axiom.
A: The Axiom of Choice is a classic example of the danger of "intuition". The quip by Jerry Bona, Professor of Mathematics at the University of Illinois at Chicago, seems appropriate here: 

"The Axiom of Choice is obviously true, the well-ordering principle
  obviously false, and who can tell about Zorn's lemma?" — Jerry
  Bona

The irony is that all three of these principles can be proven mathematically equivalent so if you accept Choice as "obvious" then you must also accept the counter-intuitive well-ordering principle and Zorn's lemma, as well as the Banach–Tarski paradox mention by a commenter. 
