Changing colours in amigurumi is really very simple. The most important thing to remember is that the colour change is worked at the end of the previous stitch.
Work the last stitch before the colour change as usual but don’t complete the stitch. Stop when you have 2 loops on your hook.
Yarn over in the new yarn colour and draw through those 2 loops.
Colour change complete. The next stitches will now be in the new colour.
How you deal with the tail ends is completely up to you. My personal preference is to cut and tie each one as I think it gives a cleaner finish but it is quite time consuming. It also depends on what sort of colour changes you’re doing, for example: if you’re switching back and forth between colours for stripes you may not need to tie off at all as you can simply carry the yarn up the inside of the piece.