For copying existing groups, in weight paint mode, press the space bar and type "Copy vertex group", and hit enter:
A new group will appear under "Vertex Groups" under the "Object data" tab. Next, hit space bar again and type "Mirror vertex group", and hit enter:
Now, from the list of "Vertex Groups", select "Group", and press Tab to enter Edit Mode. Deselect all vertices by pressing the A key until the vertices aren't highlighted. Under "Vertex Groups", four new buttons have appeared. With "Group" selected, click "Select". This will select the vertices in this group:
Next, from the list of "Vertex Groups", select "Group_copy", and click Select. Now you should have four vertices selected in total:
Next, press the + button to add a new vertex group:
With the new vertex group selected, click "Assign":
Now, if you tab back into Weight Paint mode, selecting the various groups from the list of Vertex Groups will show the different vertex weights assigned to each group
For mirroring while painting, ensure that "X Mirror" is enabled under the "Options" tab of the tool panel: