When working with solid core wire, one of the easiest ways to connect two wires together is interlocking J hooks.
Strip the wire insulation equally on both wires
Using a pair of needle-nose pliers, come down a quarter of the way on the exposed wire and bend the wire around to form a "J" shape. Repeat this step on the second wire.
Cut a piece of heat shrink tubing that will cover the exposed wires and a bit of insulation on each side. Slide the tubing down one of the wires for later.
Bring the two wires together by hooking the ends on each other.
Press the ends of the wire down with your fingers to pliers to form a solid mechanical connection.
Solder the connection
Slide the heat shrink over the newly soldered connection so it is completely covered.
Using a lighter or heat gun, shrink the tubing down on the connection.