"Carrying Yarn Up the Side" is a posting I made on this Thread and the answer I got was wonderful!
You add the colors at the beginning of the row by starting the 1st stitch, but grabbing the new yarn instead of the previous yarn and then complete the stitch. There's probably a video you can watch on that. It's really simple. Then I usually make a temporary knot until it's time to carry it up or weave it in, whichever way you choose.