Oh no, I hope everyone is okay! The first and easiest way is to immediately exchange information at the scene after the accident.
Obviously, this isn’t possible once you leave the scene so the next best way is calling the police. Many states require you to call immediately if the accident results in personal injuries, fatalities, or at least $1,000 in damages, so you may have to call them anyway. The police will have to file a report, which should have the other driver’s insurance information. Since you were involved in the accident, you can ask for a copy of the report.
If this isn’t an option, you can check with the DMV or your insurance company. Generally, the DMV is an easy place to get the information you need as they will have a record of the driver’s registration. You’ll need to prove a legitimate reason for your inquiry. Similarly, if you only have partial information about their insurance, your insurance company may be able to help.