Is this worth fighting. Yes. Will it affect his insurance? Yes unless you get the traffic school to acknowledge responsibility. How much should I go after the driving school? A lot. They obviously gave you an incompetent and dangerous instructor.
Did he hit the other car? What are the damages? Did the an Officer observe the event?
Assuming no damage to any other vehicle, If you plan on fighting this yourself (pro se) then go for it, you don't have much to lose except for court costs. The prosecutor might negotiate a favorable deal. If you plan on hiring an attorney, I would say that it is not worth it. This is because what you pay an attorney will probably outweigh the cost of the ticket and the increase in your insurance premiums combined (again, I'm assuming no damages).
As far as the driving school, read your contract with them carefully. There may be clauses that limit or immunize their liability.
BTW, If it's worth fighting, expect your insurance company to get involved.
*DISCLAIMER* This answer is for general informational purposes only, and is not intended as legal advice, nor does this constitute an attorney-client relationship in anyway.