Which is the best, clean way to join wire together
I would solder the wires together and then use heatshrink tubing to cover the joint and help make things look neat and tidy. 

Not sure how much you know, so I'll just keep going...

Solder is made to join electrical contacts, and it makes a durable bond. You need a soldering iron to heat solder. The basic approach is to hold the iron on the two (or more) wires to join, then apply solder to the heated joint. The solder will melt quickly and harden even faster after you take the iron off. That's a super basic description, and there's a ton of soldering tutorials around the web you can look to if you need help.

Heatshrink sounds like what you want if electric tape is "too messy". You basically cover the joined wires with it, then use hot air (via hot-air rework station, or just a hair drier) to shrink the tubing around the wires.