I am looking for a general way of determining cosets for $(G\times H)/H$, where $G$ and $H$ are Lie groups.
For example what are the cosets $(SU(3)\times SU(2))/SU(2)$. Is there a general method of determining it? (I am actually trying to use it to find the triviality of a fiber bundle whose base space is Grassmann and fiber is $O(n)$.)
The way the question is phrased is a little ambiguous. How does $H$ sit inside $G\times H$ as a subgroup? If it sits inside it in the canonical way as $\{1\}\times H$, then the space of cosets is canonically isomorphic to $G$ and each coset is simply $G \times \{g\}$ for $g$ an element of $H$. I.e., for each element of H there is a different coset. There is nothing to do.
Now if $H$ sits inside a little differently, as it might in your example, since $H\subseteq G$ also, the concrete forms of the cosets will differ. But the picture will look the same, the coset space will be isomorphic to the above, it's just that the cosets will concretely be different.
The main issue is whether you have $H$ sitting inside $G\times H$ as a normal subgroup or not. In the first case above, it is normal, and the coset space happens to be a group itself. But if you have put SU(2) inside of SU(3) in any of the infinitely many diferent ways, then it is not a normal subgroup.