Describe explicitly $\mathbb{Z}[\sqrt{2},\sqrt{3}]$ The question is if the following is correct, and if it's redundant or it is easier than what I've done.
I'm using this  definition of ring adjunction in Bill Dubuque answer (which is the usual used for example, also here.)
Then $\mathbb{Z}[\sqrt{2},\sqrt{3}] := R$ it's the minimal ring (minimal by inclusion obviously, the smallest possible) (inside the ring extension $\mathbb{R}$ containing both $\mathbb{Z}$, $\sqrt{2}$ and $\sqrt{3}$. Well, using this answer, I will try to extend it to two variables. I can say  that one explicit description is all polynomials over $\mathbb{Z}$, on two variables X and Y, with X evaluated in  $\sqrt{2}$, $\sqrt{3}$.
Now, that explicit description is the minimal subring we are searching, because
1) It is a subring since it is closed under product (product of polynomials evaluated in the same parameters in the evaluation of the product) and also closed by addition (by the same argument as product) and also contains inverses for addition (negating the coefficients of the polynomial, they are still in $\mathbb{Z}$ and they yield minus the evaluation of the non-negated), in the exact same way as detailed in the answer that inspired this. 
2) Any ring that contains this elements, must contain products and sum of elements of $\mathbb{Z}$ $\sqrt{2}$ $\sqrt{3}$, and thus those polynomials, being sums and products of this things, must be inside.
Now, instead of looking at ALL the evaluations, we can use the fact that $(\sqrt{2})^2=2$ wich is in $\mathbb{Z}$, and the same for $(\sqrt{3})^2=3$ wich is also in $\mathbb{Z}$. Then, that means that when evaluating the polynomials, we don't need exponents larger than 1 for X or Y, since they will return integers, so no "new" number will come out of that.

If you have $\sqrt{2}$ and $\sqrt{3}$ in a subring of $\mathbb{C}$, then also $\sqrt{6}$ belongs to it.
Now, the set
$$
\{a+b\sqrt{2}+c\sqrt{3}+d\sqrt{6}:a,b,c,d\in\mathbb{Z}\}
$$
is a subring of $\mathbb{C}$ containing $\sqrt{2}$ and $\sqrt{3}$. Hence…
An alternative, but not so explicit, description is: take all elements of the form $P(\sqrt{2},\sqrt{3})$, for $P\in\mathbb{Z}[X,Y]$.
