When you're ready to start working the second color, simply drop the first color and pick up the yarn for the second color. Knit the first stitch.
This stitch will look really loopy, loose and awful when you first knit it, but persist. When you're further along in the row you can tighten up the last stitch of the previous row and the first stitch on this row by pulling gently on the yarn tails.
Speaking of yarn tails, make sure you leave a tail of at least six inches at each end of each stripe. This will make it easy to weave your ends in securely later.