i just finished a few illustrations with a similar technique. It's not difficult - I recommend not using a plugin or a program.
In illustrator:
Start out by picking 4-6 basic colors (from darkest to lightest in your photo) Make them swatches. Create a new layer on top of your placed image. Start tracing the complete shape of your dog (darkest color). It helps to reduce the transparency to 40%. When finished, create a new layer on top, and start tracing the next set of shapes that fall in a similar color group. Keep repeating until done.
That's awesome...nice work! Can I ask what tablet you have/use? This is something I would also like to get into and tracing with the pen tool alone in ai. is very tedious.