how do I move a set of meshes to a different layer?
You hold Shift in order to multi-select many objects. After you selected all your objects, you can move them to a new layer by pressing M to select the layer you want them to be in.
Or use M folowed by the layer number, for example: M3 will move the selected objects to layer 3
There is an alternative method for moving one object at a time from one layer to another than the method detailed by WhovianBron3. Select one of the objects you want to move, open the object context button of the properties editor window, and in the "Relations" pane (illustrated below),
select the layer(s) in which you want to appear.
With both this method, and the one by WhovianBron3, the same object can be assigned to appear in multiple layers at the same time. One difference between the two methods is that the mesh can be reassigned in the properties editor window while in edit mode; to use the M key shortcut, one must be in object mode.