First of all, get acquainted with the provisions of your insurance policy - now, not after a need arises. Each company, and possibly each policy with a company, can have its own rules for submitting a claim. To get paid quickly, follow the rules!
One example of this is the need to get more than one quote for repairs on a vehicle you have insured. Some companies require two while others may ask for three estimates. Some also designate a place to get an estimate as well. Usually they will approve the lowest bid for the work.
Occasionally a step must be completed before the estimate-getting step. The insurer may want their own claim adjuster, a person who's seen a lot of banged up vehicles, to look at yours. The adjuster will see your vehicle at the tow site, if it's stuck there. However, if you're able to drive it, the company will probably have you meet the adjuster in the parking lot of their office building for a look see.
These days more often than you might imagine, the cost to repair a vehicle may exceed what that machine could be sold for on the open market, that is, its value. In that case, the vehicle is labeled as "totaled." That doesn't mean you won't fix it; it just means the insurance company is limited by two things: the value of the auto PLUS the limit of the policy you've been paying on all these years.
If you can, part of filing or submitting a car insurance claim should begin right after the accident. You should take pictures of the site and of the vehicles involved. Of course, you should get the insurance information and contact information of the other vehicle involved and you need to file a police report where this is required.
One thing that's helpful is a drawing of the layout of the accident. Don't go all Rembrandt, but grab the notepad out of the glove box (there just in case) and sketch the scene. If you've got a cell phone that takes photos - and you know how to use it - snap some pictures from every pertinent angle. Or run for a disposable at a nearby store.
What if you're at fault? Whether you are or not, don't go off blathering about it. When you must refer to the accident, try to state facts without embellishing them with statements of fault. It's also usually a better idea to keep your reports and opinions between you and your insurance company. No need talking freely with the other party or the other party's insurer.
By having accurate information and by doing exactly what your insurance carrier requires, submitting a car insurance claim does not have to be too frightening. The best plan is to read your policy thoroughly so that you know both your rights and your responsibilities when you actually file a claim.