I would recommend against butt splicing, wire nutting, or twisting and electrical taping the connections...all of those leave room for connections to work themselves loose and you can end up with weird electrical problems down the road...especially butt splicing since if they don't crimp just right, the wires either won't have a solid connection or they'll pull right out.
IMO...go to Radioshack and pick up one of their basic solder iron kits...they're like $10 and include solder, a heat sink and a few others things.
Also, pick up a pack of shrink tubing sized according to the wires in your harness.
Then solder all your wires together...make sure you put the shrink tubing on the wire BEFORE you solder it together. The shrink tubing will shrink best with the use of a heat gun (or maybe even a hair dryer), but if you don't have either handy...you can usually lightly run the tip of the solder iron across it and make it shrink down. That will help ensure, good, solid connections.