Most people don't think to carry a camera with them everywhere they go, but they will bring a cell phone, so make sure your's is one with a good camera, so you can take pictures which may become very important evidence later.
Yes the driver, because some of them have a tendency to disappear after an accident. It may make them angry, but do it casually as you take pictures of the damage to the vehicles, and the exact positioning of the cars, and don't risk getting into a confrontation with the other driver if they become angry (in that case call 911 immediately). And get a photo of the license plate in case any of the other information turns out to be false. Also, snap photos of any cars parked or stopped in the area, they may belong to witnesses.
You can move your vehicle out of traffic if no one needs emergency medical attention, but don't leave the scene until you have exchanged insurance information with the other driver.
Record the other driver's license number, insurance company name and policy number, and driver's name, mailing address, and phone number.
Witnesses can become very important in proving, on a more likely than not basis, who is at fault for the accident, and they disappear if no one explains to them just how important they can become in the future.
The sooner you have a written version of how you remember the accident, the more reliable your version of the story becomes. By the way, that is true for witness memories also. Ask them to put what they remember into writing as soon as possible too.
Injuries don't appear until 2-3 days after an accident sometimes. And, simply not feeling any pain, doesn't mean you are not injured.
Insurance companies expect you to report an accident within a reasonable period of time after it happens. If you don't, they have the right to claim you have materially breached your obligations under the contract and to deny coverage. If the accident is not your fault, the company will usually NOT hold it against you.
Although most insurance agencies will send out an adjuster to calculate the damage to your car, it's always a good idea to get a second opinion for the person of your choosing.
After a serious accident, the frustration of paying medical bills will start immediately after arriving to the hospital if not sooner, an knowing that you have good car insurance coverages to handle those expenses will enormously reduce the stress that is inevitably going to happen if your physical abilities become limited or challenging because of the accident. Talk to your insurance agent about these coverages.