I have a grid with some of its cell types as grid (subgrid). I would like to traverse the subgrid by using the up and down arrow keys and then select a row with the mouse or by using the tab key. Right now this is not happening on my subgrid. How can i fix this problem?

SubGrids are separate objects, which not included in grid hierarchy, so native navigation will not include them
You can check treegrid object which more suitable for described behavior, in case of dhtmlxgrid it will require a pretty complex code to implement required behaviour