If you believe the adds... start with one company, switch to another to save a few hundred bucks, then switch to another for a few hundred more, etc... eventually you'll end up at your original company paying nothing!
i have never been able to find a new company that will give me a better rate. most of the time they say i'll save $400 and there always 200+ more than my current ins.
My advice is to shop for company reputation rather than price. The company that charges the least is the company most likely to refuse to pay when you have a claim, or to try to low-ball your claim. I have all my car and home insurance with State Farm, and I've always been happy with them. They are probably not the cheapest, but they've always treated me right, and that's worth more to me than a few bucks difference in premium.
We had a couple accidents (were rear-ended in two different vehicles) in 2007. In both cases the other insurance company was State Farm. I was very impressed with how they paid up without weaseling! (Payments were timely and medical was asked about more than once) (We use Farmers Insurance and never had any problems with them paying up for us.)