How to Change Colors
Switching colors in crochet works best when you do it at the very end of the final stitch in the old color. Handling it this way keeps the color change looking cleaner and more natural.
In our example below, the last stitch on the row is worked in dark green, and the plan is to begin the next row in a lighter green.
On that final stitch, insert the hook as you normally would and draw up a loop — but this is where we pause and don't finish the stitch off in dark green.

To bring in the lighter green, take the light green yarn and fold it over to form a sort of fake loop. Slip the hook into this loop while holding both ends of the new strand, since it isn't anchored to anything yet. Pull that light green loop through both dark green loops on the hook.
With the color now swapped, give both the dark green and light green tails a gentle tug to neaten and tighten the stitch.
Next, we'll work the first stitch in the new color — this is what locks the new yarn in place. At this point, the light green yarn running to the ball becomes your working yarn. Hold the other tail of the new color, along with the old working yarn, behind the next stitch.
Insert the hook into the next stitch, catching both yarn tails (the old one and the new one) as you go. You'll wind up with 4 yarn strands on the hook on top of the original loop.
From here, yarn over and pull through all 4 strands from the tip of the hook. Then yarn over once more and pull the new yarn through both loops on the hook. That secures the new yarn, leaving you with a single loop on the hook in the fresh color.
After a few more stitches, you can tuck the short tail of the new color toward the inside of the amigurumi. If you intend to switch back to the old color later, keep that yarn carried along in the work; if not, you can trim it off as well.
For amigurumi, color changes tend to look their best when you take it slow and keep your tension even, since every stitch is on full display. On flat pieces, the very same method works just as nicely and keeps the transition smooth.