How to quickly determine which number is bigger than the other? $$\{1,2,66,99\}\cup\{5,7,9\}=\{1,2,5,7,9,66,99\}$$
But if I have for example 
$$\{0,2,2\sqrt{2},2\sqrt{3}\}\cup\{\sqrt{5},\sqrt{7},2\sqrt{9}\}$$
Should I first find whose element is bigger than the other and only then I can find the answer?

A set is just a collection of mathematical objects, it doesn't matter if they aren't well ordered. Imagine you have a bag full of oranges, of different sizes, is it necessary to arrange those oranges by size in order for it to be called a collection of oranges?
And so, you can simply write: $$\{0,2,2\sqrt{2},2\sqrt{3}\}\cup\{\sqrt{5},\sqrt{7},2\sqrt{9}\}=\{0,2,2\sqrt{2},2\sqrt{3},\sqrt{5},\sqrt{7},2\sqrt{9}\}$$ just as you can write it this way: $$\{\sqrt{5},2,2\sqrt{3},2\sqrt{2},\sqrt{7},2\sqrt{9},0\}.$$
