Surfaces with canonical sheaf which is trivial in some power There is a notion of Enriques surface, which is a quotient of a $K3$ surface by a group of order $2$, and thus its canonical line bundle is non-trivial but has trivial square.
My question is about some similar examples. Namely, how to construct surfaces $X_1,X_2,X_3$ such that
1) $K_{X_1}\neq 0$ but $3K_{X_1}=0$,
2) $2K_{X_2}\neq 0$ but $4K_{X_2}=0$,
3) $2K_{X_3}\neq 0$, $3K_{X_3} \neq 0$ but $6K_{X_3}=0$?
 A: First note that if $K_X^n$ is trivial for some $n \geq 0$, then for every $k \geq 0$, 
$$P_{nk} = \dim H^0(X, K_X^{nk}) = \dim H^0(X, \mathcal{O}_X^k) = \dim H^0(X, \mathcal{O}_X) = \dim \mathcal{O}(X) = \dim \mathbb{C} = 1$$ 
so $\operatorname{Kod}(X) = 0$. 
Note that $0 = c_1(\mathcal{O}_X) = c_1(K_X^n) = nc_1(K_X)$, so $c_1(X) = -c_1(K_X)$ is $n$-torsion. It follows that $X$ must be minimal (the exceptional divisor of a blow-up is not torsion). 
In the book Compact Complex Surfaces by Barth, Hulek, Peters and Van de Ven, there is a table to summarise the Kodaira-Enriques classification of compact complex surfaces, together with some additional information about certain invariants; see Chapter VI, table $10$. The relevant part of the table is below.
$$
\begin{array}{|l|c|}
\hline
\text{Class of}\ X & \text{smallest}\ n > 0\ \text{with}\ K_X^n = \mathcal{O}_X \\
\hline
4)\ \text{Enriques surfaces} & 2\\
5)\ \text{bi-elliptic surfaces} & 2, 3, 4, 6\\
6)\ \text{Kodaira surfaces} & \\
\ \ \ \ a)\ \text{primary} & 1\\
\ \ \ \ b)\ \text{secondary} & 2, 3, 4, 6\\
7)\ K3\ \text{surfaces} & 1\\
8)\ \text{tori} & 1\\
\hline
\end{array}
$$ 
So the examples that you are looking for can occur as bi-elliptic surfaces or secondary Kodaira surfaces. These surfaces are described in Chapter V, section $5$ of the aforementioned book.
In the bi-elliptic case, $X$ is of the form $(E\times C)/G$ where $E$ and $C$ are elliptic curves, and $G$ is a finite subgroup of translations on $C$ which acts on $E$ not by translations. In terms of constructing an $X$ with $K_X^n = 0$ for certain $n$, which elliptic curve is chosen to be $C$ makes no difference, but the choice of $E$ does. This is because $G$ is not acting by translations on $E$ and not all elliptic curves have the same automorphism group modulo translations. More precisely, the automorphisms of $\mathbb{C}/\Lambda$ modulo translations is $\mathbb{Z}_6$ for $\Lambda = \mathbb{Z}\oplus\mathbb{Z}\omega$ where $\omega$ is a primitive cube root of unity, $\mathbb{Z}_4$ if $\Lambda = \mathbb{Z}\oplus\mathbb{Z}i$, or $\mathbb{Z}_2$ otherwise. 
Choosing $E = \mathbb{C}/\mathbb{Z}\oplus\mathbb{Z}\omega$, one can arrange for $K_X$ to be $3$- or $6$-torsion, and choosing $E = \mathbb{C}/\mathbb{Z}\oplus\mathbb{Z}i$, one can arrange for $K_X$ to be $4$-torsion. A full list of possible groups $G$, together with their actions on $E$ and the corresponding order of $K_X$, can be found on page $199$. Alternatively, you can see it here.
