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Just about any soldering iron should work. Chisel tips are better than round tips for most work. I really like J tips as well, they've got a range of usable surface sizes without having to change tip just by turning the iron around.

Put a piece of heat shrink tubing on one wire.
Strip the ends, and form a Western Union splice in the wires to hold them.
Set the iron to 350°C, and let it heat up. If your iron doesn't have temperature control, it's cheap crap and should probably just be thrown in the trash since it'll tend to over-heat and lift pads when soldering PCBs. Continue for now, that doesn't matter as much for soldering wires.
Then apply a tiny bit of solder to the tip of the iron so that it can make good contact, apply flux to the bit where the wires join (do NOT skip flux), touch the solder to the wires, and then touch the iron to the other side of the wires. The solder should quickly melt & flow into the joint.
Remove the solder, then remove the iron.
Let the joint cool, then slide the heat shrink up over the joint and shrink it with a heat gun.

Crimp connectors tend to be stronger and more vibration-proof than solder, but sometimes space constraints mean that soldered splices are necessary. Also crimpers are expensive, for many wire-to-wire crimp families the official crimpers are several hundred dollars.