I don't go to the trouble of braiding lots of tiny filaments. I kind of flatten the wires out, lay them on top of each other and twist.
The stripped ends need to be long enough that you can twist and they lock together. That way the connection has some mechanical integrity. Then solder them. I use a mini-butane torch for 14 gauge and up. Heavy wires will suck all the heat right out of a lightweight soldering iron tip. The tip must have more mass than what you're soldering.
I use rosin-core electronic solder for everything, the kind you can buy on spools from Radio Shack or any electronics supplier.
