What is the proper way to solder two different gauge stranded wires?
If the wires are for low voltage application such as connecting 12V up to LED Strip lighting then you should consider just using twist on wire connectors instead of soldering. These have a decided advantage in that they can be removed for future wire changes and they provide an insulated cover over the wire join. 

To keep strain off the low voltage wiring connections you can zip tie the two wire cables together over the cable jacket a couple of inches from the end of the jackets.