In Blender 2.8 i have been unable to find the location of the subdivide button in any menu in edit mode and have only been able to get it using the search key. The lack of any keybind is also making it quite difficult. I am using beta 2.80.41
The following info are based on 2.8 and onward. 2.79 and previous versions have it differently.
By default, the fastest way to subdivide between edges selection is Right-Click, followed by pressing S. Right-Click is now a fully integrated context menu (its content is based on the active menu and the type of selection) You can see the Subdivide in the Context Menu when in Edit Mode with Edge active.
If you don't Right-Click after selecting the edges and then press S you'll activate the selection scaling mode. If that happen, just right-click to cancel and turn-off the selection scaling mode.
If you have the Right-Click set to Select (which was the standard in 2.79 or previous), then you won't be able to bring up the menu with that key. In that case, you can press W to call the Vertex Context Menu (Subdivide is the first item in this menu)
You can only subdivide once now. To get more than one subdivision you need to use the Loop Cut tool (Ctrl + R ), unless you want to subdivide something like a cone.