Write down the information that is accessible to you such as the car license number without asking the driver, this will show you know what you’re doing and also save time. This also means if the driver runs off you can still find out their details, after obtaining the information you can see start to ask the driver for other information.
Ask for some witness statements, it’s a good idea to get their contact details in case there are any discrepancies in yours and the other parties chain of events – this is very important as the police don’t always get it right and sometimes the at fault driver may be a good friend or acquaintance of the police officer and the accident could be rewritten to make you at fault (this is rare, but it has and does happen so it is best to protect yourself by having witnesses whom can tell the truth if needed).
You should then take photos of the accident scene and the damage that has been inflicted on both vehicles (this is why its good to have a camera phone, digital camera or even a disposable camera in your car at all times along with a cell phone). If there are any skid marks, or anything else that may be relevant then make sure you document it with photos.
As soon as possible file an accident report, and make sure you obtain a copy when it becomes readily available. Sometimes police are reluctant to attend accidents that do not have any injuries, you should insist that an officer comes, this way you are guaranteed to get any information that the driver was previously unwilling to share. And you are better protected and covered.