Changing yarn colors within a row can be neat and easy. The trick is to prepare for the color change while working the stitch before the one you want in the new color.
Put another way, the new color is introduced on the final yarn over of the stitch before.
For this project, I have worked my round so far in a cream color and am ready to switch to another color. For my last stitch with cream, I start my double crochet as usual—insert hook where indicated, yarn over and pull up a loop. Then yarn over and pull through 2 loops—but for that final yarn over of my stitch, I yarn over and pull the new color through the last 2 loops to complete my double crochet.
Now continue with the new color as indicated in the pattern. If you are going to be switching back to the first color after a bit, you can carry the yarn across the row working over it with your stitches.
If you are done with the first color (or if it is not used again for quite a while), you can fasten it off and weave it in.
Keep on Happily Crocheting,
Happily Hooked Team