I've been having a blast learning to crochet, but have found some gaps in the information available to newbies like me. My (three!) How-To guides all show how to change colours in the middle of a row while working double crochet back and forth, but not how to change colours in a new round of a granny square or other motif shape.
After some playing around and experimenting, I found a way that works for me without leaving a bulky knot hanging over an open space in the previous row.
After you've finished off the yarn from the last round, you're ready to begin the new round with the new colour. In this case, I'm about to be working [ 5 dc, ch 3, 5 dc ] into the chain space in the corner.
Pass the hook through the space from the front. Fold the end of the new yarn over the hook, leaving six inches for sewing / weaving in the end of the yarn (Photo 1). Pull the loop through the chain space (Photo 2).
Wrap both the working yarn and the end over the hook (Photo 3), and pull them both through the loop, together (Photo 4). The stitch you just made anchors the end of the yarn in place, but you still need to weave the cut end in later (once you have enough completed stitches in that colour through which to weave and hide the end). I made the next chain stitch with just the working yarn (Photo 5), and you can see the slightly thicker stitch made with the doubled yarn.
From then on, just crochet as normal. In the case of this square, I worked enough chain stitches to be equivalent to a double crochet, then worked the remaining stitches that were needed in the space.