Beside transcendental or uncomputable numbers what other types of numbers are there? What other types/categories of numbers are there that we know of today (i.e. some one has done some work on them, like Chaitin's uncomputable $\Omega$ number)?
Of course there are uncountably many types/categories of numbers (besides the trivial integer, rational, natural numbers, ...) due to the uncountability of the reals.
Are there any generalisations of irrationality/transcendentality for reals by analogy to complex numbers (2-tuples) or $n$-tuples? What I am trying to ask is: are the reals the only playground for number categorisation (such as irrationality, trancendentality, ...) ? or does anyone know of some other structure similar to $\mathbb R$ (by some type of analogy!) with analogous irrationality/algebraicness/transcendentality properties?
 A: You might be interested in learning about the $p$-adic numbers. 
A: The Euclidean Constructible Numbers are of great historical importance.  Roughly speaking, these are the numbers that we can produce, starting from $0$ and $1$, by using the ordinary arithmetical operations (addition, subtraction, multiplication, and division) as well as taking the square roots of non-negative numbers.
Thus, for example,
$$\frac{\sqrt{10-3\sqrt{2}}}{17}$$
is Euclidean constructible.
The reason they are interesting is that they are closely connected to the problem of what are the possible geometric constructions by straightedge and compass.  For example, it was proved (possibly by Gauss, certainly by Wantzel) that the number $\cos(20^\circ)$ is not Euclidean constructible.  This was the key step in showing that the famous ancient problem of trisecting the general angle cannot be done in general by straightedge and compass.
At the same time, Wantzel proved that the number $\sqrt[3]{2}$ is not Euclidean constructible. This shows that the old problem of Duplicating the Cube is not solvable by straightedge and compass. (You are given the side of a cube, and you want to construct the side of a cube which has twice the volume of the given cube.)
Some thirty years earlier, Gauss had proved that the regular $17$-gon is straightedge and compass constructible, basically by showing that the number $\cos(360^\circ/17)$ belongs to the Euclidean constructible class of numbers.
This was the first new result about Euclidean constructibility of regular polygons since ancient Greek times.  Supposedly, it was finding this result that made Gauss decide to concentrate on mathematics, and not philology.
Another historically very important class of numbers are the (complex) numbers that are obtainable by using the ordinary operations of arithmetic, together with $n$-th roots for arbitrary integers $n$.  Understanding this class of numbers was a key idea in the proof by Galois that there is no general "formula" for solving equations of degree $5$. (About $300$ years earlier, it had been shown by Cardano, and others, that there is a general formula for solving equations of degrees $3$ and $4$).  Of course people had known how to solve quadratic equations for far longer than that.
And then there are the endlessly many extensions of the notion of number.  You might find the Surreal Numbers of Conway interesting. 
A: Besides the classes mentioned so far, "there exist" hyperreal numbers which can involve infinite and infinitesimal quantities, interval numbers, and fuzzy numbers which come in a countable infinity of types themselves.
For the last question, when asking about structure, what operations or relations do you pair up with R?
A: Related: I would recommend a good read, Numbers by Ebbinghaus et. al. It is a very interesting book on the history and development regarding numbers.
