Here's the technique you need when your knitting pattern calls for a change of color.
Drop the color you've been working with behind your work. Insert the RHN into the next stitch, then pick up and lay the old color on top of the RHN. Let it lie there as you pick up the new color with your left hand, and wrap it under and around the needle as if you were going to knit the stitch. Before completing the knit stitch, push the old color off the top of the needle, then complete the knit stitch with the new color.