You can move a selected layer by grabbing and moving it around with the mouse. To restrict movement to a particular axis, hold down ⇧ when you drag.
As you move (or resize) your layer, Sketch should automatically align your layers to adjacent layers. If this doesn’t happen, you might have accidentally switched off the Smart Guides. To turn them on, go to View › Canvas › Show Smart Guides in the menu. Rulers, guides and grids are quite a big topic so they have a chapter of their own.
If you hold down the ⌥ key while dragging, you will clone the layer, leaving the original layer in its place. If you then immediately press ⌘ D (duplicate), Sketch will insert another duplicated layer at exactly the same offset as your first duplicate.
Moving an Obscured Layer
Another complication of overlapping layers is moving an obscured layer. Normally, when you click and drag a shape it will immediately be selected - if it wasn’t selected yet - and moved to its new location.
Very often this is a convenience, but it can be a hindrance as well if you’ve got a layer selected that is completely obscured by another layer. Clicking on the layer would instead select the layer on top and move that one.
To get around this, you can hold hold down the ⌥ ⌘ keys and then clicking and dragging will not change your selection. You can even click and drag somewhere completely different on the canvas and Sketch will still preserve your selection:
Note: In Sketch 2.x this behaviour was using the A key instead but since version 3, the A key has been remapped to insert an Artboard instead.