Changing Colors Between Rounds:
You can transition from one color to the next between rounds. Work in the old color right up until the very last step of the last stitch in the round. Then pull a loop of the new color through to complete the last step. Cut the old color, leaving a long tail of yarn, and continue crocheting with the new color.
There are multiple ways to attach new yarn colors. This suggestion above is one way of doing it, but there are other ways that would work just as well.
Joining new yarn creates loose ends. Whenever possible, work overtop of these loose ends to hide them. If you are not familiar with how to do that, you can see a separate tutorial for that on this page: How to crochet over yarn ends.
See more on page 4: Instructions for crocheting round 2 of the granny square.