If you are buying a second car, your no claims discount may only apply to your first car, but you may still get a lower car insurance quote if you have not made a claim previously and use that car insurance company for your first car.
Although it's cheaper to have third party fire and theft, if you drive a more expensive car make a comprehensive policy may offer more protection for any damage to your vehicle. Many companies do not allow third party insurance for cars over a specific value for your protection.
If you've had an accident don't get any repair work done until you have cleared it with your car insurance company first. It is possible to choose a car insurance company who have agreed car repairers, to save you driving around getting quotes. They normally gaurantee their work also.