Is there any way to select touching/joined vertices/edges/faces in edit mode?
There are several ways:
Select at least one element (face/vert/etc.) and press CtrlL to Select Linked.
Press L while hovering over an element with the mouse to select all elements connected to that element.
Select at least one element and press CtrlNumpad + to Grow the selection until everything is selected.
Select at least one face and press CtrlShiftAltF to select connected faces which have the same or similar normal/direction.
If you meant you want to select all elements that are connected to other elements, create/select a single vertex and press ShiftG> Amount of connecting edges (if you want don't want to select edges that don't connect to faces, use Amount of adjacent faces instead), then press CtrlI to invert the selection.
See the wiki for a full list and description of all selection tools.
Press B for border select, or C for circle select, or hold and drag LMB while holding Ctrl to make a lasso select.
P.S.: For overlapped elements (if it is what you mean), It should be done in Wireframe display mode, or toggle off the Limit selection to visible button on the 3D View header.