To move the whole group you can use the "Move tool" it's found in the same area as the pick tool. Just click on the white arrow next to the pick tool and it'll reveal the move tool.
The move tool allows you to move any layer, as well as move all layers that that layer is linked to. Creating links to layers allows you to define certain layers as needing to belong to each other. You will use the move tool specifically to move these guys together.
Layer groups automatically link all of the layers within the group together for you so that you don't have to worry about doing this yourself. You can unlink layers within the layer group by selecting the "link/unlink group" button found on the layers palette.
It should be warned, however, that the move tool is a little finicky. It'll select the topmost layer that the cursor is clicking on. So if you have a mask layer this is considered covering whole image and trying to click on ti will select the mask layer. So to avoid that you'll have to turn off any mask layers while using the move tool.