cyclic group definition i would like to clarify some things related to group theory,for example let us consider two group,one with modulo $3$ and second  with modulo $2$,so we have
$G_3=(0,1,2)$  and $h_2=(0,1)$
now if i consider  sum of this group,for example let us take  some elements
$(0,1)+(1,2)$
i should know  with modulo is taken first and which second ?i was watching Youtube  video  and professor  said that
$(0,1)+(1,2)=(1,0)$
also one example
$g=(1,1)$
$g+g=(1,1)+(1,1)=(0,2)$
why it is $(0,2)$?  zero means that  $2\ ({\rm modulo}\ 2)=0$ and  $2$ means simple $2\ ({\rm modulo}\ 3)$,but how  should i understand which one is which?

Let's look at the given cyclic groups:
$H = \{0, 1\},\;$ under addition modulo $2,\;$ and $\; G = \{0, 1, 2\}\;$ under addition modulo $3$.
Then $$H \times G = \{(h, g)\mid h\in H, g \in G\}, \\ \text{ And }\;(h_1, g_1) + (h_2, g_2) = \Big((h_1 + h_2) \text{ mod } 2,\;(g_1 + g_2) \text { mod } 3\Big)$$

Note that the order of groups in the direct sum does matter: 
$$G\times H = \{(g, h)\mid g\in G, h \in H\},\\\text{ and }\;(g_1, h_1) + (g_2, h_2) = \Big((g_1 + g_2) \text{ mod } 3,\;(h_1 + h_2) \text { mod } 2\Big)$$

In the examples you give, it is clear that the direct sum of the groups is $H \times G$.
Note that if we have an element $(1, 1) \in H\times G$, while you name it $g$, it is NOT an element of G; it is an element in $H\times G$, represented as an ordered pair, with the first of the two entries is an element of $H$, and the second of the two entries is an element of $G$. 
So $(1, 1) + (1, 1) = \Big((1 + 1) \text{ mod }2,\; (1 + 1) \text{ mod } 3\Big) = (0, 2)$.
